Question

test for convergence : summation 3power-n-(-1)power-n​

Answer 1

Answer:

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Step-by-step explanation:

The series converges to 1/2.

The partial sums of the are 1/3, 4/9, 13/27, 40/81 …

So the sum from 1 to k appears to equal (3^n -1) /(2 * 3^n)

We can prove this by induction: for n = 1 you get (3^1–1)/(2* 3^1) = 2/ (2*3) = 1/3

assume k: ( 3^k - 1)/(2* 3^k)

For k+1: ( 3^k - 1)/(2* 3^k) + 1/3^(k+1) = 3*( 3^k - 1) +2/(2* 3^k) *3 = 3^(k+1) -3+2/283^(k+1) = 3^(k+1)-1/(2*3^(k+1) which is what we needed to show to prove this.

So the limit as n →infinity of (3^n-1)/(3^n *2) = 1/2 since limit of (3^n -1)/3^n = 1

so 1 * 1/2 = 1/2

Answer 2

Step-by-step explanation:

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