# Calculus - series

19,658 results

### Calculus

By recognizing each series below as a Taylor series evaluated at a particular value of x, find the sum of each convergent series. A) 1+5 + (5^2)/(2!)+(5^3)/(3!)+(5^4)/(4!)+...+ (5^k)/(k!)+...= B) 1-(2^2)/(2!)+(2^4)/(4!)-(2^6)/(6!)+...+((-1)^(k)2^(2k))/((2k)!) +...=

### Integral Calculus - Series

Find if series is convergent or divergent. Series from n=2 to infinity (4n+7)/(3n^3 -8n)

### Calculus

a) Find the Taylor series associated to f(x) = x^-2 at a = 1. Be sure to show the general term of the series. b) Find the radius of convergence of the series. c)Use Lagrange's Remainder Theorem to prove that for x in the interval of convergence with x > 1; the power series ...

### Calculus

Determine the following about the series. Indicate the test that was used and justify your answer. Sigma (lower index n = 1; upper index infinity) [sin((2n-1)pi/2)]/n A. The series diverges B. The series converges conditionally. C. The series converges absolutely. D. It cannot...

### Calculus

Consider the infinite series of the form: (+/-)3(+/-)1(+/-)(1/3)(+/-)(1/9)(+/-)(1/27)(+/-)...(+/-)(1/3^n)(+/-)... (A) Find x and y from: x(</=)(+/-)3(+/-)1(+/-)(1/3)(+/-)...(</=)y. (B) Can you choose the signs to make the series diverge? (C) Can you choose the signs to ...

### calculus

Consider the infinite series of the form: (+/-)3(+/-)1(+/-)(1/3)(+/-)(1/9)(+/-)(1/27)(+/-)...(+/-)(1/3^n)(+/-)... (A) Find x and y from: x(</=)(+/-)3(+/-)1(+/-)(1/3)(+/-)...(</=)y. (B) Can you choose the signs to make the series diverge? (C) Can you choose the signs to ...

### calculus

Consider the infinite series of the form: (+/-)3(+/-)1(+/-)(1/3)(+/-)(1/9)(+/-)(1/27)(+/-)...(+/-)(1/3^n)(+/-)... (A) Find x and y from: x(</=)(+/-)3(+/-)1(+/-)(1/3)(+/-)...(</=)y. (B) Can you choose the signs to make the series diverge? (C) Can you choose the signs to ...

### Calculus

How can I prove this series alternating series converges(this is the answer)? (-1)^2*(2/3)^n I tried it this way: an = (2/3)^n, then i just broke it down. 2^n/(3^n) and i took the ratio of it and got 2/3 which does not equal to one which would mean the series diverges.. but ...

### College Calculus

Suppose the series An (from n=1 to INF) is known to be convergent. Prove that series 1/(An) (from n=1 to INF) is a divergent series. I have no idea what to do... please help!

### College Calculus (Binomial Series)

Expand f(x) = (x+x^2)/((1-x)^3) as a power series and use it to find the sum of series (SUM from n=1 to infinity) (n^2)/(2^n) PLEASE HELP.

### calculus

With power series, is an endpoint convergent if you plug it back into the original series, and get an alternating series that is conditionally convergent?

### Calculus

Find a power series, centered @ x=0, for function f(x)=x/(1+2x). I know this is a maclaurin series, but my work doesn't get the right answer. Can you please show steps? Also,do all power series start with a 1, as in (1+2x+4x^2+...)? Thanks in advance!

### Calculus

Find a power series, centered @ x=0, for function f(x)=x/(1+2x). I know this is a maclaurin series, but my work doesn't get the right answer. Can you please show steps? Also,do all power series start with a 1, as in (1+2x+4x^2+...)? Thanks in advance!

### Pre-Calculus

Q.Determine the sum of each infinite geometric series. t_1= 8 r = -2^1/2 ---------------------------------------- A.This is a divergent series because the absolute value of r is greater than 1. ---------------------------------------- Q. The first term of an infinite geometric...

### Integral Calculus

We can use this power series to approximate the constant pi: arctan(x) = (summation from n = 1 to infinity) of ((-1)^n * x^(2n+1))/(2n+1) a) First evaluate arctan(1) without the given series. (I know this is pi/4) b) Use your answer from part (a) and the power series to find a...

### calculus

Consider ∞ ∑ [(3k+5)/(k²-2k)]ᵖ, for each p ∈ ℝ. k=3 Show this series { converges if p > 1 { diverges if p ≤ 1 Hint: Determine the known series whose terms past the second give an approximate match for the terms of this series. This ...

### Calculus

Obtain the MacLaurin series for 1/(2-x) by making an appropriate substitution into the MacLaurin series for 1/(1-x). ------------ The MacLaurin series for 1/(1-x) = Σ x^k I substitue (x-1) in for x, because 1/(2-x) = 1/(1-(x-1)) Making the same substitution in the ...

### Calculus

Hi, could someone please help me with this question? Thanks Calculate the Maclaurin series of f(x)=4x^3*sin(x^4). Explicitly use the Maclaurin series for sine. Using this series for f(x), verify that the integral of 4x^3*sin(x^4)dx=-cos(x^4)+C.

### calculus

in the following series x is a real number. In each case use the ratio test to determine the radius of convergence of the series. Analyze the behavior of the series at the endpoints in order to determine the interval of convergence. A) (nx^n)/(n^2 + 2) B)((n^2)(x-2)^n)/2^n C...

### Calculus - Taylor #2

Find the Taylor series for f(x) centered at the given value of 'a'. (Assume that 'f' has a power series expansion. Do not show that Rn(x)-->0.) f(x) = x3, a = -1 and what i've done so far: f (x) = x^3 f ' (x) = 3x^2 f '' (x) = 6x^1 f ''' (x) = 6x f (-1) = -1 f ' (-1) = 3 f...

### CALCULUS-URGENT

find the radius and interval of convergence for the series the series from n=0 to infinity of ((-1)^n*x^n)/(n+1)

### CALCULUS

find the radius and interval of convergence for the series the series from n=0 to infinity of ((-1)^n*x^n)/(n+1)

### Calculus

does infinite power series -1/n diverge? (i think so, because it is just the negative of the harmonic series)...? thank you!

### Calculus 2

Hello, I don't know what test to use for this series: Determine the sum of the following series: inf E n=1 (2^n + 9^n) / 12^n thank you!

### CALCULUS-URGENT- no one will respond!!!

find the radius and interval of convergence for the series the series from n=0 to infinity of ((-1)^n*x^n)/(n+1)

### Calculus, series

I cannot figure this out! What is the series (the Pattern) of this sequence? {1,5,1,5,1,5,1,5.......}

### calculus

how do you find the sum of a series? for example: the series from n=0 to infinity of (-1)^n/n!??? thanks

### calculus

Use division of power series to find the first three terms of the Maclaurin series for y = sec x.

### calculus

I need help understanding how the series of e derives into the exponential series using the binomial theorem. Here is a link to a pic of a page in my book, regarding the exponential series: ht tp://i46.tiny pic(.)(com)/qz0oat . jpg (remove parentheses and spaces) A couple of ...

### calculus

state the power series of an appropriate familiar function and use it to calculate the power series of the given function. give answer in sigma notation. All power series have center at 0. f(x)=(e^(-x)^2) -1+(x^2) and g(x)=x cos (x/square root 3)

### Calculus

Show that the following series is absolutely convergent: Summation from 1 to infinity: [(-1)^n * (n+1) * 3^n]/ [2^(2n+1)] I've done the ratio test and replaced n in this series with n+1. I get 3/4 in the end, which is less than 1, which confirms that the series is abs. ...

### calculus

i have to determine whether the series is convergent, and if, find the sum the series is from k=1 to infinity of 2/((k+1)(k+3)) I got 5/6 as my answer and didn't know if it was right...

### calculus

what does this series, if it converges, converge to? the series from n=1 to infinity of (2^n + (-1)^n)/3^n

### calculus

Use multiplication of power series to find the first three non-zero terms of the Maclaurin series of e^x ln(1 − x).

### Calculus

The divergence test applied to the series ∑n=1 to ∞ 3n/(8n+9) tells us that the series converges or diverges? I got that it was divergent because it was undefined at infinity, is my answer right?

### Calculus

If S1 = 0.7 and S2 = 2.1 in geometric series, what would the sum of the first 12 terms in the series be? I tried doing this, and I got 1.16 or something :S How exactly do I do this? Please tell me what formula to use. I was using the Sn = a(rn-1)/r-1

### Calculus - - Reiny

If S1 = 0.7 and S2 = 2.1 in geometric series, what would the sum of the first 12 terms in the series be? I tried doing this, and I got 1.16 or something :S How exactly do I do this? Please tell me what formula to use. I was using the Sn = a(rn-1)/r-1

### Calculus 2 (Series)

Can anyone help me start this problem from beginning to end, along with explanations on how to go about the problem for a better understanding how to do this series problem? 1) Find the values of p for which the series is convergent. Summation notation symbol (with n=1 on ...

### calculus

test the series for convergence or divergence the series from n=0 to infinity of (x^2+1)/(x^3+1) I said that due to the limit comparison test this converges at 1

### calculus

test the series for convergence or divergence the series from n=0 to infinity of sin (2^-x) I wasn't sure what test to use to see if this was or wasn't convergent

### calculus

determine whether the series is convergent if so find sum it is the sum from k=1 to infinity of ((-1)^k)/(3^(k+1)) i found this series to be geometric where a=-1/9 and r=1/3 my answer was converges to 1/6

### calculus

test the series for convergence or divergence using the alternating series test the sum from n=1 to infinity of (-1)^n/(3n+1) I said it converges, is this true?

### calculus

test the series for convergence or divergence using the alternating series test the sum from n=1 to infinity of (-1)^n/(3n+1) I said it converges, is this true?

### calculus

Show that the series(−1)^n−1(bn) where bn = 1/(n^1/2) if n is odd and bn = 1/2^n if n is even, diverges. Why does the alternating series test fail?

### calculus

Another problem: determine whether the series is convergent if so find sum it is the sum from k=1 to infinity of ((-1)^k)/(3^(k+1)) i found this series to be geometric where a=-1/9 and r=1/3 my answer was converges to 1/6

### Pre-calculus

Which of the following series is divergent? a) 1+3(1/4)+9(1/4)^2+27(1/4)^3... b) 1+3(1/5)+9(1/5)^2+27(1/5)^3... c) 1+3(1/7)+9(1/7)^2+27(1/7)^3... d) 1+3(1/2)+9(1/2)^2+27(1/2)^3... How do you determine if a series in convergent or divergent??? The book that I have is about as ...

### Calculus

If you have a geometric alternating series, and you prove that the series is converging by doing geometric series test, and NOT alternating series test, then does that allow you to say that the series converges ABSOLUTELY? Or should you do alternate series test also to say ...

### calculus

for each series determine if the series is absolutely convergent and convergent the sum from 0 to infinity of (-1)^n/(the square root of (n+1))

### calculus

test the series for convergence or divergence the series from n=1 to infinity of 1/(arctan(2n)) I again didn't know what test to use

### calculus

does this series converge, and if so is it absolutely convergent? the series from n=1 to infinity of ((-1)^*n+1))/n^4 I found that by the ratio test it was inconclusive, so no abs. conv is this right? and how do i know if it is simply convergent?

### calculus

does this series converge, and if so is it absolutely convergent? the series from n=1 to infinity of ((-1)^*n+1))/n^4 I found that by the ratio test it was inconclusive, so no abs. conv is this right? and how do i know if it is simply convergent?

### Calculus 2

The following function has a series of the form the sum from n=0 to infinity of c(subn)x^n. Calculate the coefficients c(subn) and express the power series in summation notation. f(x)=(pi*x)/(pi*x+1) Thank you so much for your help!!!!

### Calculus

Consider an infinite series of the form (+-)3(+-)1(+-)1/3(+-)1/9(+-)1/27(+-)....(+-)1/3^n(+-)... The number 3,1, etc. are given but you will decide what the signs should be. a)Can you choose the signs to make the series diverge? B)Can you choose the signs to make the series ...

### Calculus Derivative- Taylor Series?

let f(x)= x/x-1 find f'(x) f ''(x) and a formula for f ^ (n) * x. I found the first and second derivatives but do not know how to make a general equation for this. I have not learnt the Taylor or Maclaurin Series either. Thank you.

### calculus

Where did the exponential series come from? 1 + x + x^2/2! + x^3/3!... Where did that number come from? and how is it used to get the trigonometric series?

### calculus

How would you determine the power series of 1/(1-x)^3. I know that the series of 1/(1-x) is x^n, but how would you manipulate it for this scenario?

### calculus - power series ASAP please :)

using power series, integrate & evaluate to 4 dec. places integral from 0 to 1: sin x^2 dx i'm REALLY stuck on this. and i need help asap.. what is the inverse of "sin x^2" so that i could have it in a fraction that will fit the power series equation? and that is: (A)/(1-R) ...

### Math

Consider the following series: 1,2,3,4,5,10,20,40....which starts as an arithmetic series? ...but after the first five terms becomes a geometric series. Prove that any positive integer can be written as a sum of distinct numbers from the series. I know how to do the base case

### Excel Help

3) __________ are used to compare sets of data in one chart. Time series Multiple data series Relative series Comparison series Is multiple data series

### calculus

use the power series to estimate the series: from 0 to 4 of ln(1+x)dx with absolute value of the error less than .0001/ Give your estimate of the integral as well as a bound on the error. I found the 'terms' in the series to be: x-(1/2)x^2+(1/3)x^3-(1/4)x^4...... with a radius...

### calculus

use the power series to estimate the series: from 0 to 4 of ln(1+x)dx with absolute value of the error less than .0001/ Give your estimate of the integral as well as a bound on the error. I found the 'terms' in the series to be: x-(1/2)x^2+(1/3)x^3-(1/4)x^4...... with a radius...

### Calculus 2 (Series - Convergent or Divergent?)

Can someone show me a step by step process and explanation how to solve this problem? 1) Consider the following series. (∞ on top of summation symbol) (k = 1 under the summation symbol) ∑ k(k+15)/(k+13)^2 Determine whether the series is convergent or divergent. If ...

### calculus

How do you differentiate the maclaurin series for 1/(1-x) twice to find the maclaurin series of 1/(1-x)^3.

### Calculus

This is going to be pretty hard to show as text since it would be easier for me to post a picture of the question. The question has f(x) = x/4^2 - (2x^3)/4^4 + (3x^5)/4^6 + ... . I am trying to find out the value of f(2). There is a hint to differentiate the power series ...

### calculus

determine whether the series converges of diverges the sum from k=2 to infinity of (the square root of (ln(k)))/k I said that because you can't integrate the series (goes to infinity) it diverges

### calculus

determine whether the series converges of diverges the sum from k=2 to infinity of (the square root of (ln(k)))/k I said that because you can't integrate the series (goes to infinity) it diverges is this true?

### calculus

determine if the series is absolutely convergent and convergent the sum from n=1 to infinity of sin(n^2)/n^2 what series test should I use and how? the ratio test?

### CALCULUS

find the radius and interval of convergence for the series the series from n=1 to infinity of ((-1)^(n+1)*x^n)/n! I did the ratio test so I had the Lim as n approaches infinity of -x/(n+1), but this is 0, giving no radius, so I think I did something wrong...

### Algebra 2

I need steps on how to complete this please i am so confused and lost. :( Consider the infinite geometric series x e n=1 -4(1/3) n-1. In this image, the lower limit of the summation notation is "n=1". a. Write the first four terms of the series. b. Does the series diverge of ...

### CALCULUS-URGENT

find the radius and interval of convergence for the series the series from n=1 to infinity of ((-1)^(n+1)*x^n)/n! I did the ratio test so I had the Lim as n approaches infinity of -x/(n+1), but this is 0, giving no radius, so I think I did something wrong...

### Calculus 2

Find the radius of convergence and interval of convergence of the series. n=2 series to infinity (-1)^n * x^n+6/n+7 R= ? I= (or[ , ]or) How do i do this?? ( means convergent and [ means divergent.

### Integral Calculus

For what values of p is this series convergent? (summation from n = 1 to infinity) of ((-1)^(n-1))/(n^(p + 2)) A. p >= -2 B. p =/= -2 C. p > -2 D. for all p E. p > 0 You have to use the Alternating Series Test. I've already tried E, and it was wrong. I have one more ...

### CALCULUS-URGENT- no one will respond!!!

find the radius and interval of convergence for the series the series from n=1 to infinity of ((-1)^(n+1)*x^n)/n! I did the ratio test so I had the Lim as n approaches infinity of -x/(n+1), but this is 0, giving no radius, so I think I did something wrong...

### Calculus - series

I'm getting this answer wrong, can someone please help show me what i'm missing?? thank you :) Infinity of the summation n=0: [(-1)^n pi^(2n)] / [6^(2n) (2n)!] this is my work: [(-1^0) pi^(2*0)] / [6^(2*0) (2*0)!] + [(-1^1) pi^(2*1)] / [6^(2*1) (2*1)!] + [(-1^2) pi^(2*2)] / [6...

### Chemistry

Hydrogen exhibits several series of line spectra in different spectral regions. For example the Lyman series (nf = 1 in Balmer-Rydberg equation) occurs in the ultraviolet region while the Balmer (nf = 2) series occurs in the visible range and the Paschen (nf = 3), Brackett (nf...

### Calculus Help Please

Hello! I am struggling with this problem: Find the Taylor Series for sin(x-2) centered at c=3. My work so far: sin(1)+(x-3)cos(1)-(1/2)(x-3)^2sin(1)-(1/6)(x-3)^3cos(1)+(1/24)(x-3)^4sin(1)... I know that the Taylor series for sign is typically: the summation from n=0 to ...

### algebra, series

The sum of the first n terms of a series is 1-(3/4). Obtain an expression for the nth term of the series. Prove that the series is geometric, and state the values of the first term and the common ratio. Please show workings

### math

for this infinite series (-1)^n/n^2 if i use alternating series test to show that sequence does of a^n does not go to 0, then does this mean that this series is diverging

### Arithmetic Series - - Reiny

Consider the series defined by Sn = 3n-1 Find the first four terms of the series. How exactly do I do this?

### math

hi if i wanna find the Maclaurin series from another existing series by integrating that series , do i need to include the constant of integration ?? Thanks

### calculus

determine whether the series converges of diverges the sum from n=1 to infinity of 1/(the square root of (n^3+1)) I said that through the comparision test (comparing to 1/the square root of (n^3) the series converges is this true?

### Calculus

Can someone prove (informally) the following theory: If there is a differentiable function, f, that is represented by a Taylor series, T, then the convergence interval for series T is identical to the convergence interval for the term-by-term derivative T'.

### calculus

determine whether the series converges of diverges the sum from n=1 to infinity of 1/(the square root of (n^3+1)) I said that through the comparision test (comparing to 1/the square root of (n^3) the series converges

### calculus

is this correct? use the integral test to determine if this series is convergent or divergent: the series from n=2 to infinity of 1/(n*square root of (ln(n))) I said it was divergent because the integral went to infinity

### math

the last two tems in a geometric series are 1080 and 6480 and the sum ofthe series is 7775 what is the first term in the series

### Calculus

In calculus, the sum of an infinite geometric series whose first term is 1/20 is given by the complex fraction 1/20 _____ 1-r where r is the common ratio between the terms. Simplify this expression.

### chemistry

Hydrogen exhibits several series of line spectra in different spectral regions. For example the Lyman series (nf = 1 in Balmer-Rydberg equation) occurs in the ultraviolet region while the Balmer (nf = 2) series occurs in the visible range and the Paschen (nf = 3), Brackett (nf...

### Induction

Given the series 1,2,3,4,5,10,20,40,80,..... which starts as an arithmetic series for the first five terms and then follows geometric series. It can be proved by strong induction that any positive integer can be written as a sum of distinct numbers of this series. Has to be ...

### Calculus!!!

I need your help... I've posted this question a while ago but no one has answered it yet. Please please help me. Which of the following series can be used to compute ln(.8)? a) ln(x-1) expanded about x=0 b) lnx about x=0 c) ln x in powers of (x-1) d) ln(x-1) in powers of (x-1...

### Chemistry

Hydrogen exhibits several series of line spectra in different spectral regions. For example the Lyman series (nf = 1 in Balmer-Rydberg equation) occurs in the ultraviolet region while the Balmer (nf = 2) series occurs in the visible range and the Paschen (nf = 3), Brackett (nf...

### Calculus III

Which of the following series are geometric series? Find the sum if they are 1. Infinity (Summation sign) n = 1 1/6n^2 2. Infinity (Summation sign) n = 1 (0.6)^n-1

### calculus

for each series determine if the series is absolutely convergent and convergent the sum from 0 to infinity of (-1)^n/(the square root of (n+1)) I did the ratio test and got -1, which is less than 0 making it absolutely convergent, but do i need to take the absolute value of -1...

### Calculus II

(a) Use differentiation to find power series representation for f(x)=1/(1+x)^2 What is the radius of convergence? (b) Use part (a) to find a power series for f(x)=1/(1+x)^3 (c) Use part (b) to find a power series for f(x)=x^2/(1+x)^3 I found part (a) which was Σ n=0 to &#...

### calculus

Approximate the value of the series to within an error of at most 10^−5. The infinite series from n=1 to infinity (−1)^(n+1)/(n^6) according to the equation (2): |S_n - S| is less than or equal to a_(n+1) what is the smallest value of N that approximates S to within an ...

### calculus repost, please assist pleassssssse

Hi Trying to work with confusing Taylor series....any assistance would be much appreciated!! How can I use T(x)=5-9((x-2)^2)-3((x-2)^3) to approximate the f(0) ???? I realize that Taylor series f(x)=f(a)+f'(a)(x-a)+f''(a)((x-a)^2)etc... but the above T(x) is centered at x=2, ...

### Math (College Level Mathematics)

Fourier sin series for f(x) = 1, 0 < x < Pie is given by 1 = 4/n E 1/ (2n-1) times sin (2n-1) x, (0 < x < n). Using this, find the Fourier sinc series for f(x)= 1, on 0 < x < c where c > 0. Then find the Fourier series for g(x), x > 0 where g(x) = 1, 0...

### Calculus 2

Calculate c(sub0), c(sub1), c(sub2), c(sub3), and c(sub4) for the power series the sum from n=0 to infinity of c(subn)x^n that represents f(x)=tanx. Use the first two nonzero terms of the series to approximate the tangent of 1/4 radian. Compare your approximation with the ...

### Calculus

how do I know if the series ln(n)/(2n) is DV or CV?

### Calculus II

Does anyone know of a very good website that can teach me to better understand the disk, washer, shell method to finding volume of irregular figures (solid of revolution)using of course, integration techniques (such as when to recognize integration with respect to y or X for ...