Question

determine the limit of the integral [(3x-1)/(2n+3)]^(2x) from 1 to infinity, using the root test.

Answer 1

is that series? integral ? what ?

i guess

\[\large \lim_{n \rightarrow \infty} (\frac{3n-1}{2n+3})^{2n}\]

Answer 2

the sum from n=1 to infinity (or the integral both ways the limit is the same)

Answer 3

i got it

Answer 4

but yes we are talking about finding the sum of a series using the root test

Answer 5

well Using root test

\[a_{n}= (\frac{3n-1}{2n+3})^{2n}\]

\[\large \lim_{n \rightarrow \infty} (a_{n})^{\frac{1}{n}}=\lim_{n \rightarrow \infty} [(\frac{3n-1}{2n+3})^{2n}]^{\frac{1}{n}}=\lim_{n \rightarrow \infty} (\frac{3n-1}{2n+3})^{2}=\frac{9}{4} > 1\]

Answer 6

since we know that

\[\lim_{n \rightarrow \infty} \frac{3n-1}{2n+3}=\frac{3}{2}\]

so the series

\[\large \sum_{n=1}^{\infty} (\frac{3n-1}{2n+3})^{2n}\]

diverges

Answer 7

ok thanks! i just forgot to square the answer i was putting in 3/2

Answer 8

very well :)