# Solve Sigma 1/(k(k+1)): Step-by-Step Guide

• MHB
• ChelseaL
In summary, the conversation discusses using the fact that \frac{1}{k} - \frac{1}{k+1} = \frac{1}{k(k+1)} to show that the sum of \frac{1}{k(k+1)} from k=1 to n is equal to 1-\frac{1}{n+1}. The method involves breaking the sum into two parts and using bracketing to simplify the expression. The final result is S_n = \frac{n}{n+1}.
ChelseaL
Use the fact that $$\frac{1}{k}$$ -$$\frac{1}{k+1}$$ = $$\frac{1}{k(k+1)}$$ to show that

n
sigma ($$\frac{1}{k(k+1)}$$) = 1-$$\frac{1}{n+1}$$
r=1

What do I need to do to solve it?

What they want you to do is:

$$\displaystyle S_n=\sum_{k=1}^n\left(\frac{1}{k(k+1)}\right)=\sum_{k=1}^n\left(\frac{1}{k}\right)-\sum_{k=1}^n\left(\frac{1}{k+1}\right)=\sum_{k=1}^n\left(\frac{1}{k}\right)-\sum_{k=2}^{n+1}\left(\frac{1}{k}\right)$$

Now, take off the first term of the first sum, and the last term of the second sum like so:

$$\displaystyle S_n=1+\sum_{k=2}^n\left(\frac{1}{k}\right)-\sum_{k=2}^{n}\left(\frac{1}{k}\right)-\frac{1}{n+1}$$

What are you left with?

Sn = 1 + (1/n+1)?

ChelseaL said:
Sn = 1 + (1/n+1)?

To properly use bracketing, you want:

Sn = 1 + 1/(n+1)

I would highly recommend learning to use $\LaTeX$ to make your expressions more readable. :)

However, that is incorrect, as you've got the wrong sign in front of the second term. What you want is then:

$$\displaystyle S_n=1-\frac{1}{n+1}$$

This is what we've been asked to show. I would choose to write it this way though:

$$\displaystyle S_n=\frac{n}{n+1}$$

Thank you!

## What is Sigma notation?

Sigma notation is a mathematical notation used to represent a sum of terms. It is written as Σ (capital sigma) followed by the expression or formula for the terms, with the starting value on the bottom and the ending value on the top of the sigma symbol.

## What does "1/(k(k+1))" represent in the given expression?

This expression represents the general term of the given series. In other words, it is the formula for each term in the series, where k is the index or position of the term in the series.

## How do I solve a series in Sigma notation?

To solve a series in Sigma notation, you need to follow these steps:

1. Start by writing out the general term of the series, as given in the expression.
2. Substitute the values of the index (k) for each term in the series.
3. Add the resulting terms together.

## What is the purpose of the "Step-by-Step Guide" for solving Sigma notation?

The step-by-step guide is designed to help you understand the process of solving a series in Sigma notation. It breaks down the steps and provides a clear explanation of each step, making it easier for you to follow along and solve the series correctly.

## Why is it important to know how to solve Sigma notation?

Sigma notation is commonly used in mathematics and science to represent sums of terms. Knowing how to solve it allows you to calculate the sum of a series, which can be helpful in various real-world applications, such as finance, physics, and statistics.

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