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...

  1. 👍
  2. 👎
  3. 👁
  1. There are many tests for convergence. In this case you can compare the series to the known convergent series 1/k^2 from k = 1 to infinity. Since
    2/((k+1)(k+3)) is always less than 2/k^2 the series must converge.

    The summation can be computed by writing:

    2/((k+1)(k+3)) =

    1/(k+1) - 1/(k+3)

    So, you would expect the summation to be given by the sum of 1/(k+1) from k = 1 to k = 2 (the two terms that are not conceled by 1/(k+3)), which is the answer you obtained.

    Since the two separate summations do not converge you must prove that this is indeed the case. What you do is you say that the summation from 1 to infinity of 2/((k+1)(k+3)) equals the limit of N goes to infinity of the summation from 1 to N. For finite N you then evaluate the summation of

    1/(k+1) - 1/(k+3)

    from k = 1 to N.

    You then get four terms: the k = 1 and k = 2 terms of 1/(k+1) which yields 5/6 and the k = N-1 and k = N terms of
    - 1/(k+3). So, you get:

    5/6 - 1/(N+2) - 1/(N+3)

    Finally, you take the limit
    N --> infinity which yields the answer of 5/6

    1. 👍
    2. 👎
  2. The terms can be rewritten
    1/(k+1) -1/(k+3)
    I used the technique of partial sums. You can verify that the two expressions are equivalent.
    Summing from x=1 to infinity, all terms will cancel in pairs except 1/2 + 1/3, for k=1 and k=2. The sum of those terms is 5/6, so I agree.

    Did you use a different method?

    1. 👍
    2. 👎
  3. kinda helpful

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. 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)

  2. Calculus

    Which one of the following statements is true about the series the series from n equals 1 to infinity of the quotient of negative 1 raised to the nth power and n ? (4 points) Is this A or B? I am a little confused. A) It is

  3. Calculus

    The series the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and n is convergent. Use the Alternating Series Test to find an upper bound on the absolute error if the 7th partial

  4. Calculus

    How many terms of the series the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and n do we need to add in order to find the sum with an absolute value of its error to be less than

  1. calculusII

    Determine whether the series is convergent or divergent.∑from k=1to ∞ (ke^-9k)

  2. Calculus

    Use the alternating series test to determine the convergence/divergence of the series the summation from n equals 1 to infinity of the product of negative 1 raised to the nth power and the quotient of 3 times n and the quantity 4

  3. Calc 2

    Determine whether the series is convergent or divergent series symbol n=1 to infinity (n^2/(e^(3n))

  4. Calculus - Alternating Series Test

    Determine whether the infinite series, sigma(((-1)^(n+1))/n)^2 converges or diverges. My professor gave these in a problem set after he taught the alternating series test. Simplying the series we get, sigma(((-1)^(n+1))/n)^2

  1. algebra

    Consider the infinite geometric series below. a. Write the first 4 terms of the series b. Does the series diverge or converge? c. If the series has a sum, find the sum. ∞ ∑ (-2)^n-1 n=2 not sure how to do this at all

  2. Calculus

    Given that the series the summation (from n = 1 to infinity) ((-1)^(n + 1))/n is convergent, find a value of n for which the nth partial sum is guaranteed to approximate the sum of the series with an error of less than 0.0001. 9

  3. Pre Cal.

    Find three geometric means betwee -sqrt(2) and -4sqrt(2). A: 2, -2sqrt(2), 4 Is 6 sqrt(2)+ 6 + 3sqrt(2)+... a convergent series? A: Yes. Is 1 + 3(1/2) + 9(1/2)^2 + 27(1/2)^3 +... a divergent series? A: Yes. Are these right?

  4. 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

You can view more similar questions or ask a new question.