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^(2*2) (2*2)!] + [(-1^3) pi^(2*3)] / [6^(2*3) (2*3)!]
-1 + -0.13707783 + -0.00313172 + -0.0000286 + -0.00000014
sum of the series = -1.14023829
You got the signs wrong. The answer is 1/2 sqrt[3]
thank you!!!
To find the correct sum of the series, let's break it down step by step.
First, let's rewrite the series in a more general form:
∑((-1)^n * pi^(2n)) / (6^(2n) * (2n)!), where n starts from 0 and goes to infinity.
Now, let's analyze the individual terms in the series.
The numerator, (-1)^n * pi^(2n), represents a pattern where the exponent of (-1) alternates between positive and negative values as n increases. The base, pi, remains the same for all terms, and it is raised to the power of (2n), which doubles as n increases.
The denominator, 6^(2n) * (2n)!, also follows a pattern. The base, 6, remains the same for all terms, and it is raised to the power of (2n), which doubles as n increases. The factorial of (2n) represents multiplying all the integers consecutively from 1 to (2n).
Now, let's simplify the individual terms of the series based on the given formula.
For the first term, when n is 0:
[(-1)^0 * pi^(2*0)] / [6^(2*0) * (2*0)!]
= 1 / (1 * 1)
= 1
For the second term, when n is 1:
[(-1)^1 * pi^(2*1)] / [6^(2*1) * (2*1)!]
= -pi^2 / (36 * 2)
= -pi^2 / 72
For the third term, when n is 2:
[(-1)^2 * pi^(2*2)] / [6^(2*2) * (2*2)!]
= pi^4 / (216 * 24)
= pi^4 / 5184
Now, let's add up all these terms:
1 + (-pi^2 / 72) + (pi^4 / 5184)
To add these terms together, we need a common denominator. In this case, the lowest common denominator is 5184. Let's multiply each term by different forms of 1 to obtain that common denominator:
(5184/5184) + (-72pi^2 / 5184) + (pi^4 / 5184)
Now, combine the terms:
(5184 - 72pi^2 + pi^4) / 5184
Since we're dealing with an infinite series, we need to evaluate whether this series converges (reaches a finite value) or diverges (goes to infinity).
Based on the simplified form above, since the exponent of pi in the numerator is even, and the exponent of 6 in the denominator is also even, the series converges. Let's find the actual sum:
The simplified form becomes:
(5184 - 72pi^2 + pi^4) / 5184
We can factor this expression as follows:
( (pi^2)^2 - 72(pi^2) + 5184 ) / 5184
Now, let's find the roots of the quadratic equation formed within the numerator:
(pi^2 - 36)^2 = 0
Taking the square root of both sides:
pi^2 - 36 = 0
pi^2 = 36
pi = ±6
Since we are working with the positive value of pi in our series, we can conclude that pi = 6.
Substituting pi = 6 back into ( (pi^2)^2 - 72(pi^2) + 5184 ) / 5184:
( (6^2)^2 - 72(6^2) + 5184 ) / 5184
= ( 1296 - 2592 + 5184 ) / 5184
= 1
Therefore, the sum of the series is 1.
Your answer of -1.14023829 is incorrect.
However, you mentioned the correct answer is "1/2 sqrt[3]." To confirm this, we need to double-check if there was an algebraic error when simplifying the series.
Please provide the steps you followed to arrive at "1/2 sqrt[3]" as the answer, and we can compare it with the result we obtained.