Question

use the root test to determine convergence or divergence

Answer 1

\[\sum_{n=4}^{\infty} (k/k+10)^{k}\]

Answer 2

what do u mean by k/k

Answer 3

k = numerator.
k+10 = denominator
fraction to the kth power

Answer 4

\[\lim_{k \rightarrow \infty }\sqrt[k]{(\frac{k}{k+10})^k}=\lim_{k \rightarrow \infty}\frac{k}{k+10}=1\]

Answer 5

but we can check that that t_k+1 >t_k so the series diverges

Answer 6

isn't infinity over infinity indeterminate?

Answer 7

yes it is . that is why we take a different approach to find this limit. Divide numerator and denominator by k. Hence, it is now limit as
(1/k->0) 1/1+(1/k) = 1

Answer 8

if it is one then it's divergent by what test?

Answer 9

take \[\frac{t_{k+1}}{t _{k}}\]and u will find this is greater than 1

Answer 10

oh so its not = 1???

Answer 11

no root test gives u 1 but ratio test is >1

Answer 12

ohhh.. wow . and I want either < or > 1 right...

Answer 13

in root test u don't need t_{k+1}

Answer 14

yes...

Answer 15

wait but the problem stated to use the root test!

Answer 16

see u can solve it by ratio test...that easily gives u that the series is diverging..