# Thread: Limit problem: lim [n -> infty] [(1 + 2^p + 3^p + ... + n^p)/(n^{p+1})]

1. ## Limit problem: lim [n -> infty] [(1 + 2^p + 3^p + ... + n^p)/(n^{p+1})]

Hello

I was wondering which areas this math problem involves in? I'm currently reading a course in calculus and I do not recognize the problem in my course literature .
Would be grateful if you could list the parts that you need to solve this problem.

. . .$\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}$

. . . . .$\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}$

2. Originally Posted by emmablom
I was wondering which areas this math problem involves in? I'm currently reading a course in calculus and I do not recognize the problem in my course literature .
Would be grateful if you could list the parts that you need to solve this problem.

. . .$\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}$

. . . . .$\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}$
What was the source of this exercise? If your textbook, then you should probably apply topics from that section, and immediately previous.

Note: Wolfram Alpha gives the limit as being zero, if 0 < p < 1. (here) So I'm guessing that you're probably expected to consider cases....

3. ## part chapter ?

The question is from a previous test in the course and I wonder what part chapter I can read in order to fully understand the problem?

This is what my course literature covers.

Iterations

Newton-Raphson's Method

Series: convergence / divergence, integral criterion

comparison criteria

root / quota criteria

Alternating series

Leibniz criterion

rearrangement of series

Power series

Taylor series

Power series convergence

Derivation / integration of power series

solution of differential equations

Sequences, series of functions

pointwise and uniform convergence

Dominated convergence

reversal of border processes.

4. Originally Posted by emmablom
The question is from a previous test in the course and I wonder what part chapter I can read in order to fully understand the problem?

. . .$\mbox{For }\, p\, >\, 0,\, \mbox{ solve the following:}$

. . . . .$\displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1\, +\, 2^p\, +\, 3^p\, +\, ...\, +\, n^p}{n^{p+1}}$

This is what my course literature covers.

Iterations

Newton-Raphson's Method

Series: convergence / divergence, integral criterion
. . . . ..comparison criteria
. . . . ..root / quota criteria
. . . . ..Alternating series
. . . . ..Leibniz criterion
. . . . ..rearrangement of series
. . . . ..Power series
. . . . ..Taylor series
. . . . ..Power series convergence
. . . . ..Derivation / integration of power series

solution of differential equations

Sequences, series of functions

pointwise and uniform convergence

Dominated convergence

reversal of border processes
I would guess that the sequences-and-series chapter(s) would be a good place to start. But you know your book better than I do. You're taking the course; where does this sort of exercise arise in the book's homework sets? Study that section, and the preceding ones.

5. thanks for the tip!