# Finding the Limit of a Series - Solve Unknown Problem

• Frillth
In summary, the conversation discusses finding the value of an infinite series using the method of taking the limit of the sequence of partial sums. The specific series in question involves finding the sum of the first n integers. The conversation also mentions a more direct approach using pictures and a formula known as "Gauss' sum". This method is also applicable to higher powers of k in the series. Overall, the conversation provides helpful tips and techniques for solving the problem at hand.
Frillth
I have the following problem:
http://img432.imageshack.us/img432/9461/problemax9.jpg
I know for a fact that the answer is not 0, but I have no idea how to actually find the answer. I've never seen a similar problem before, and I'm not really sure how to start it.

Last edited by a moderator:
Can you find a function such that if you approximate its integral over some range by n rectangular strips, you get that sum? Then the limit would just be the integral over that range.

I think there's a more direct approach.

Remember that a series is a sequence of partial sums. The general term of the sequence is

$$a_n=\sum_{k=1}^{n}\frac{k}{n^2} = \frac{1}{n^2}\sum_{k=1}^{n} k=?$$

You know how to do that sum on the left, and thus you can find an explicit form of the general term. Just take the limit.

I'm not really sure what you mean by the explicit form of the general term. Can you give me a little more help, please?

Do you know what

$$\sum_{k=1}^{n} k$$

sums to? It's sometimes called "Gauss' sum" named after the 8 years old who found the value of the sum in his head when asked to compute the sum of the first 50 integers.

So you can get that sum directly in a few different ways, the simplest of which is to consider the pictures:

*0

**0
*00

***0
**00
*000

etc.

If you know that formula, that's definitely the easiest way. What I was suggesting might be overkill, and I was thinking of it because it's the way you would go about this if you had a higher power of k, ie:

$$\sum_{k=1}^{n}\frac{k^p}{n^{p+1}}$$

Oh, duh! I can't believe I didn't think of that. Thanks!

Generally, given an infinite series

$$\sum_{k=1}^{+\infty}b_n$$

you cannot put the general term $a_n=\sum_{k=1}^n b_n$ of the sequence of partial sums in a friendly form that allows for direct computation of the value of the series by just taking the limit:

$$\lim_{n\rightarrow +\infty}a_n$$.

But the series you're dealing with is one of these rare case where the general term has a friendly form in terms of n that allows for this method of calculating the sum to work.

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## 1. What is the purpose of finding the limit of a series?

The purpose of finding the limit of a series is to determine the value that the series approaches as the number of terms increases towards infinity. This value is known as the limit and can help us understand the behavior of the series and make predictions about its future terms.

## 2. How do I solve for the unknown problem in a series?

To solve for the unknown problem in a series, you can use various techniques such as the ratio test, comparison test, or the integral test. These tests help determine the convergence or divergence of the series and can provide insights into the unknown problem.

## 3. Can I use a calculator to find the limit of a series?

In most cases, yes, you can use a calculator to find the limit of a series. However, it is essential to understand the concepts behind finding the limit and the limitations of calculators in handling complex series.

## 4. Is it necessary to find the limit of a series?

No, it is not always necessary to find the limit of a series. In some cases, it may be sufficient to determine the convergence or divergence of the series. However, finding the limit can provide more information and insights into the behavior of the series.

## 5. Are there any real-life applications of finding the limit of a series?

Yes, finding the limit of a series has various real-life applications, such as in finance, physics, and engineering. For example, in finance, the limit of a series can help predict the future value of an investment, while in physics, it can help determine the behavior of a system over time.

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