Which values for a will make series convergent or divergent?∑k=1∞1(2k+1)lna(2k+1)

This nature of this series is ruled by the so-called Bertrand criterion, which tells you that ∑n≥21nαlnβ(n) converges if and only if α&gt;1 or α=1 and β&gt;1More precisely, in your case, the general term is positive, and equivalent to ctenlna(n), so the series converges iff a&gt;1

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High School
Calculus and Analysis
Calculus 2
Series

Shyla Singleton

Answered question

2022-03-11

Which values for a will make series convergent or divergent?

\sum_{k = 1}^{\infty} \frac{1}{(2 k + 1) \ln^{a} (2 k + 1)}

Answer & Explanation

Veronica Riddle

Beginner2022-03-12Added 9 answers

This nature of this series is ruled by the so-called Bertrand criterion, which tells you that

\sum_{n \geq 2} \frac{1}{n^{α} \ln^{β} (n)}

converges if and only if

α > 1

α = 1

and

β > 1

More precisely, in your case, the general term is positive, and equivalent to

\frac{cte}{n \ln^{a} (n)}

, so the series converges iff a>1

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