1. ## Riemman Sum

The limit as x approches infinity of the sum from i=1 to n of :

$(1+\frac{i}{n})^3$

2. ## Divergent

$\lim_{n\rightarrow\infty}\sum_{i=1}^n(1+\frac{i}{n })^3$ = $\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac1{n^3}(n ^3+3n^2i+3ni^2+i^3)$= $\lim_{n\rightarrow\infty} 1+\frac3n\sum_{i=1}^ni+\frac3{n^2}\sum_{i=1}^ni^2+ \frac3{n^3}\sum_{i=1}^ni^3$ = $\lim_{n\rightarrow\infty} 1+\frac3n\frac{n(n+1)}2+\frac3{n^2}\frac{n(n+1)(2n +1)}6+\frac3{n^3}\frac{n^2(n+1)^2}4$ = $\infty$

This limit diverges. What was the original problem?
The text contains a mathematical problem involving a Riemann sum and its limit, followed by a detailed solution showing that the limit diverges.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
1. ## Riemman Sum

The limit as x approches infinity of the sum from i=1 to n of :

$(1+\frac{i}{n})^3$

2. ## Divergent

$\lim_{n\rightarrow\infty}\sum_{i=1}^n(1+\frac{i}{n })^3$ = $\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac1{n^3}(n ^3+3n^2i+3ni^2+i^3)$= $\lim_{n\rightarrow\infty} 1+\frac3n\sum_{i=1}^ni+\frac3{n^2}\sum_{i=1}^ni^2+ \frac3{n^3}\sum_{i=1}^ni^3$ = $\lim_{n\rightarrow\infty} 1+\frac3n\frac{n(n+1)}2+\frac3{n^2}\frac{n(n+1)(2n +1)}6+\frac3{n^3}\frac{n^2(n+1)^2}4$ = $\infty$

This limit diverges. What was the original problem?
user
The text contains a mathematical problem involving a Riemann sum and its limit, followed by a detailed solution showing that the limit diverges.

TYPE: SINGLE_QUESTION_WITH_ANSWER