Question

Evaluate

lim(n→∞) {{1/n [(1/n)^9+(2/n)^9+(3/n)^9+⋯+(n/n)^9 ]}.

Answer 1

\[\lim_{x \rightarrow {\infty}} \frac{(1/n)^9+(2/n)^9+(3/n)^9+⋯+(n/n)^9}{n}\]

Is this the problem?

Answer 2

this is the question

Answer 3

Is it as x approaches infinity or as n approaches infinity?

Answer 4

it is n

Answer 5

\[\frac{(1)^{9}/n^{9}+(2)^{9}/n^{9} +...+(n)^{9}/n^{9}}{n} = (1^{9}/n^{9}+2^{9}/n^{9} +...+(n)^{9}/n^{9})(\frac{1}{n})\]

\[1^{9}/n^{10}+2^{9}/n^{10} +...+(n)^{9}/n^{10} = 1^{9}/n^{10}+2^{9}/n^{10} +...+1/n\]

Answer 6

If you take the limit of the simplified fraction as n -> infinity the expression tends toward zero

Answer 7

oh wait so is that mean answer is 0?

Answer 8

Yes

Answer 9

Hmmm i was up to here

lim
n->oo { (1/n)^10 [ (1)^9+(2)^9+....+(n)^9)}

i dont get it how answer is 0

of did i went wrong way?

Answer 10

(1)^9+(2)^9+....+(n)^9 is all over n^10 and not n?

Answer 11

Ohhhh. You followed my solution up to that point?

Answer 12

i was worked on it till that part haha. and i realised

i need to find the sum formula for sum k^9 as for k = 1 to n

i should have post this in very first place but didnt confident enough..

sorry for that.

Answer 13

It's okay :)

Answer 14

Well, what I did in that step is I multiplied every number in in the parentheses

[1^9/n^9+2^9/n^9+....+n^9/n^9] by the number 1/n

which gives the result

(1^9/n^9)*(1/n)+(2^9/n^9)(1/n)+....+(n^9/n^9)*(1/n)

When you multiply all of these out you get

1^9/n^10+2^10/n^10+....+n^9/n^10

and the n^9/n^10 on the very end simplifies to 1/n which leaves us with

1^9/n^10+2^10/n^10+3^9/n^10+4^9/n^10....+1/n^3+1/n^2+1/n

and when n goes to infinity all of these values go to zero so you get

0+0+0+0+...+0+0+0 = 0

Answer 15

i was going to ask "why did u multiply by 1/n wouldnt that b changing result ?" and oh wait there is 1/n outside of breaket haha.

I GOT IT NOW HAHA THANK YOU !

Answer 16

No problem :D

Answer 17

If you need anymore help and I'm on, shoot me a @hunus