# radius of the power series

Find the radius of convergence of ∑▒〖(a_n x^n)〗, given a_n=1when n is the square of a natural number and a_n otherwise. If a_n=1 when n=m! for n∈N and a_n=0 otherwise, find the radius of convergence of ∑▒〖(a_n x^n)〗.

Determine the radius of convergence of ∑▒〖(a_n x^n)〗, if 0<ρ≤|a_n |≤q for all n∈N.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

To find the radius of the power series , we use the following theorem.

Theorem: Let , then the radius of convergence is .

Problem #1

(a) Find the radius of convergence of , given if is the square of a natural number and otherwise. I think you missed in the posting.

From the condition, we know that , then we have

So the radius of convergence is .

(b) Find the radius of convergence of , given when and otherwise.

From the condition, we know that , then we have

So the radius of convergence is .

Problem #2

Find the radius of convergence of , if for all .

From the condition, we know that , then we have

Here we use the fact that for any positive number , we have . This is because as .

Thus we must have

So the radius of convergence is .

© BrainMass Inc. brainmass.com December 24, 2021, 8:39 pm ad1c9bdddf>https://brainmass.com/math/real-analysis/radius-power-series-298876