Power series question. Calc IV final tomorrow. Help!?

[匿名]

2012-05-15 7:12 am

Sum[(n[x+2]^n)/(4^[n+1])): n goes from 1 to INF

I need to know the radius and interval of convergence. Please show steps, I have to show my work on the test.

Thanks

回答 (1)

2012-05-15 7:46 am

✔ 最佳答案

First, use the Ratio test.

r = lim(n→∞) |[(n+1) (x+2)^(n+1) / 4^(n+2)] / [n(x+2)^n / 4^(n+1)]|
..= (1/4) |x+2| * lim(n→∞) (n+1)/n
..= (1/4) |x+2|.

So, this series converges (save an endpoint or two) if r = (1/4) |x+2| < 1.
==> |x + 2| < 4. [So, R = 4.]

Next, we check the endpoints:
x = 2 ==> Σ n/4; divergent by the n-th term test (since the nth term does not converge to 0.)
x = -6 ==> Σ (-1)^n * n/4; divergent by the n-th term test (as above).

Hence, the interval of convergence is (-6, 2).

Good luck!

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