Part BCSIR NET December 2024the-radius-is-the-reciprocal-of-the-root

The radius is the reciprocal of the root

Consider the power series $\sum_{(} n = 1)^{\infty} [n^{(} n^{2}) / (n + 1)^{(} n^{2})] x^{n}$ with coefficients in real numbers $R$ . Which of the following statements is true?

A.The radius of convergence of the series is 1/e
B.The series converges at x = 5
C.The series converges at x = 3
D.The series converges for all x with |x| < 1/2✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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50 are analysed free — try those first.

Related counterexample: Σaₙxⁿ → L as x → 1⁻ ⇒ Σaₙ = L

The chapter behind this: Power series — radius, boundary, Abel — free to read

From Sequences and Series of Functions › Power series, radius of convergence, Abel's theorem

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