Question

apply the ratio test to the following series and state what you can conclude from it.

Answer 1

\[\sum_{k=1}^{\infty}\frac{ k ^{50} }{ 2^{k} }\]

Answer 2

and the ratio test:

Answer 3

\[\rho = \lim_{k \rightarrow + \infty}\frac{ u _{k+1} }{u _{k} }\]

Answer 4

Can you have a go at applying the ratio test yourself?

Answer 5

i did, just can't do the limit part

Answer 6

the series is suppose to be convergent, when i work till:\[\lim_{k \rightarrow + \infty } \frac{ (k+1)^{50} }{ 2k ^{50} }\]

i don't know how to go on.

Answer 7

ah, wait, can i use l'hospital rule?

Answer 8

Ok that's what I got also. I'm going to neglect writing the limit here but remember it's still there, I'm just lazy.
If you expanded the numerator it would look something like this:
\[\frac{ k^{50} + ... + 1 }{ k ^{50} }\]
Where all the ... terms are just multiples of k^n for n between 49 and 1.
Now if we divide by k^50 on top and bottom we get:
\[\frac{ 1+...+\frac{ 1 }{ k ^{50} } }{ 2 }\]
And for k->infinity, all of the ... and 1/k^50 will cancel.

Answer 9

(I didn't mean cancel, I mean tend to 0)

Answer 10

So can you see what we will be left with?

Answer 11

1/2

Answer 12

Indeed :)

Answer 13

thx!