(a) use differentiation to find a power series representation forf(x) =1(6 + x)2.f(x) =∞ leftparen1.gif(−1)n(n+1)xn6n+2 rightparen1.gifsum. gifn = 0what is the radius of convergence, r? r = (b) use part (a) to find a power series forf(x) =1(6 + x)3.f(x) =∞ leftparen1.gif(−1)n(n+3)(n+1)xn6n+5 rightparen1.gifsum. gifn = 0what is the radius of convergence, r? r = (c) use part (b) to find a power series forf(x) =x2(6 + x)3.f(x) =∞ leftparen1.gif(−1)n(n+2)(n+1)xn+26n +3 rightparen1.gifsum. gifn = 2what is the radius of convergence, r? r =

(a) Wild guess:

Recall the power series

With some manipulation, we can write

Take the derivative and we get

so we have

By the ratio test, this series converges if

or , so that the radius of convergence is .

(b). If we take the second derivative, we get

Apply the ratio test again and we get .

(c) Multiply the previous series by and we get

The ratio test yet again tells us .

  the correct choice is marked

step-by-step explanation:

the first two graphs show multiple turning points. the last graphs matches the description in every detail.

step-by-step explanation:

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