Question

It is useful to write series both in the form \(a_{1}+a_{2}+a_{3}+\cdots\) and in the form \(\sum_{n=1}^{\infty} a_{n} .\) Write out several terms of the following series (that is, write them in the first form). $$ \sum_{n=1}^{\infty} \frac{n}{2^{n}} $$

Short answer

\(\frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \cdots\)

Step-by-step solution

Understand the Series

This series is given in the summation notation: \ \ \ \ \ \ \ \ \ \ \(\sum_{n=1}^{\infty} \frac{n}{2^{n}}\). We need to write it in the expanded form, showing the first few terms explicitly.

Identify the General Term

The general term of the series is \ \ \ \(\frac{n}{2^{n}}\). For different values of \(n\), this term will change.

Write Several Terms

Now, substitute the first few values of \(n\) into the general term to get the series in expanded form: \ \ \ \ \[ \frac{1}{2^{1}} + \frac{2}{2^{2}} + \frac{3}{2^{3}} + \frac{4}{2^{4}} + \cdots \] \ \

Key concepts

Summation Notation

Summation notation is a compact way to represent a sum of terms. Instead of writing out each term individually, summation notation groups them into a single expression. The series \(\sum_{n=1}^{\infty} \frac{n}{2^n}\) can be interpreted as: 'sum all terms where the general term is \(\frac{n}{2^n}\), starting at \(n = 1\) and continuing to infinity.' This makes it easier to write and work with large series. Understanding summation notation is crucial for dealing with mathematical series. It simplifies complex sums, making it easier to identify patterns and compute results.

General Term Identification

The general term of a series helps to find individual terms without listing them all. In the series given by \(\sum_{n=1}^{\infty} \frac{n}{2^n}\) , the general term is \( \frac{n}{2^n}\). This formula captures the nth term in the sequence. By plugging in different values for \(n\), we can find specific terms. For example, if \(n = 1\), we calculate \(\frac{1}{2^1} = \frac{1}{2}\). If \(n = 2\), then it is \(\frac{2}{2^2} = \frac{2}{4} = \frac{1}{2}\). Understanding the general term formula is critical because it allows us to see the pattern and behavior of the series as it progresses.

Series Expansion Steps

Expanding a series means writing out its terms explicitly. Following the series \( \sum_{n=1}^{\infty} \frac{n}{2^n}\), let's break it down step-by-step:

• Step 1: Recognize the general term \(\frac{n}{2^n}\).
• Step 2: Substitute the first few values of \(n\) into the general term:

When \(n = 1\), we get \(\frac{1}{2}\).

When \(n = 2\), we get \(\frac{2}{4} = \frac{1}{2}\).

When \(n = 3\), we get \(\frac{3}{8}\).

When \(n = 4\), we get \(\frac{4}{16} = \frac{1}{4}\).



So, the expanded form of the series is:

\( \frac{1}{2} + \frac{1}{2} + \frac{3}{8} + \frac{1}{4} + \cdots\). Breaking down the series helps in better understanding how each term is formed and how the series progresses.