Question

Ratio test

Answer 1

I have this series\[\sum_{n=1}^{∞}(1-\cosh(1/n))\]
and I need to show the ratio test doesn't work i.e.
\[\lim_{n \rightarrow ∞}\frac{ a _{n+1} }{ a _{n} }=1\]
But I can't solve it, can someone help me?

Answer 2

Test? You're in a test? :p aw

Answer 3

No I using the Ratio Test to Determine if a Series Converges. :)

Answer 4

Orrh i thinke i got it.. L Hospital rule?

Answer 5

\[a_n=1-\cos h (1/n)=1-\frac{e^{1/n}+e^{-1/n}}{2}=\frac{2-e^{-1/n}-e^{1/n}}{2e^{1/n}}\]
\[a_{n+1}/a_n=?\]

Answer 6

\[a_{n+1}/a_n=\frac{ 2-ee^{1/n}-ee^{-/n} }{ 2ee^{1/n} }\frac{ 2e^{1/n} }{ 2-e^{1/n}-e^{-1/n} }=\frac{ 2-ee^{1/n}-ee^{-/n} }{ 2e-ee^{1/n}-ee^{-1/n}}\]

Answer 7

Thank you @Jonask

Answer 8

there s an error

Answer 9

i put e^1/n without having not to do so

Answer 10

downstairs i mean

Answer 11

but anyway it does not affect the final ratio we got over there

Answer 12

:)