Find all values of real number p for which the series converges absolutely sum_(k=2)^infty 1/(k(log k)^p

MISA6zh

MISA6zh

Answered question

2022-11-03

Find all values of real number p for which the series converges absolutely
k = 2 1 k ( log k ) p

Answer & Explanation

Gwendolyn Alexander

Gwendolyn Alexander

Beginner2022-11-04Added 16 answers

If we have a non-negative decreasing sequence { a n } n = 1 , 2 , , then
n = 2 a n  converges iff  n = 2 2 n a 2 n  converges
In your case, we have a n = 1 n log p ( n )
Hence,
n = 2 2 n a 2 n = n = 2 2 n 2 n ( n log 2 ) p = 1 ( log 2 ) p n = 2 1 n p
which converges for p>1 and diverges for p≤1.

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