Calculus 2 Help - Convergence/Divergence?

Question

1. What happens if a_n decreases to zero and you consider Sum  (a_n)^n?

2. Does Sum of (-1)^[n*sin(1/n^p)] converge/diverge? Prove?

3. What Power Series Representation do you get when differentiating the Power Series Representation of 1/(1-x)?

kb · Answer

1) Using the Root Test:

r = lim(n→∞) [(a_n)^n]^(1/n)

..= lim(n→∞) a_n

..= 0, by hypothesis.

Since r = 0 < 1, this series converges.

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2) Assuming that you meant (-1)^n * sin(1/n^p) for the nth term:

lim(n→∞) sin(1/n^p) = 0 <==> p > 0.

(Otherwise, the series diverges by the nth term test.)

Next, for p > 0:

(d/dn) sin(1/n^p) = (-p/n^(p+1)) cos(1/n^p) > 0 for all n > 0.

==> {sin(1/n^p)} is a decreasing sequence.

So, the given series converges by the Alternating Series Test when p > 0, and diverges otherwise.

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3) The answer is (d/dx) 1/(1-x) = 1/(1-x)^2 [within the interval of convergence (-1, 1)].

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I hope this helps!

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