if-a-n-b-n-c-n-are-real-sequence-with-a-n-gt-0-b-n-gt-0-c-n-gt-0-and-a-n-n-lt-b-n-lt-c-n-1-n-if-n-0-a-n-converge-and-n-0-c-n-converge-did-n-0-b-n-converge

Question Number 672 by 123456 last updated on 21/Feb/15

i f a_{n}, b_{n}, c_{n} a r e r e a l s e q u e n c e w i t h

a_{n} > 0, b_{n} > 0, c_{n} > 0

a n d

a_{n}^{n} < b_{n} < c_{n}^{1 / n}

i f \underset{n = 0}{\sum^{+ \infty}} a_{n} c o n v e r g e a n d \underset{n = 0}{\sum^{+ \infty}} c_{n} c o n v e r g e

d i d \underset{n = 0}{\sum^{+ \infty}} b_{n} c o n v e r g e ?

Commented by prakash jain last updated on 22/Feb/15

For \underset{n = 1}{\sum^{\infty}} a_{n}^{n}

\lim_{n \to \infty} \sqrt[n]{a_{n}^{n}} = \lim_{n \to \infty} a_{n} = 0

so this converges .

For \underset{n = 1}{\sum^{\infty}} c_{n}^{1 / n}

assume c_{n} = \frac{1}{n^{2}}

c_{n}^{1 / n} does not converge .

\underset{n = 1}{\sum^{\infty}} b_{n} cannot conclude as convergent

or divergnt .