Problem 66 - Algebra and Trigonometry

Question

Express each sum using summation notation. \(\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\cdots+\frac{n}{e^{n}}\)

Step-by-Step Solution

Answer

\(\sum_{k=1}^{n} \frac{k}{e^{k}}\)

Step 1: Identify the Pattern

Look at the terms given that follow a specific pattern. Each term in the sum can be written as \(\frac{k}{e^{k}}\), where \(k\) represents the term number, starting from 1 and going up to \(n\).

Step 2: Define the General Term

Express the general term of the given series in terms of \(k\). The general term is \(\frac{k}{e^{k}}\).

Step 3: Set Up the Summation

Now, express the entire sum using summation notation. The sum of the terms from 1 to \(n\) can be written as \(\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\cdots+\frac{n}{e^{n}} \).

Step 4: Write the Summation Expression

Using the summation notation, the sum can be expressed as \(\sum_{k=1}^{n} \frac{k}{e^{k}}\).

Key Concepts

series and sequences

Understanding series and sequences is crucial when dealing with summation notation. A sequence is a list of numbers arranged in a particular order, often defined by a specific formula. Each number in the sequence is called a term.

For example, in the sequence \[1, 2, 3, \ldots, n\], the formula for the general term (the nth term) is just \[n\]. Notice that the sequence doesn't stop at a fixed number and can go on indefinitely.

On the other hand, a series is the sum of the terms of a sequence. If you take the same sequence and add its terms, you get a series: \[1 + 2 + 3 + \ldots + n\]. This means each term in the sequence is summed up to form the series.

Understanding the difference between the two concepts helps when transitioning from sequences to series, especially when expressing these sums in summation notation.

general term

To express any sequence or series, you need to identify its general term. The general term, often denoted as \(a_k\), represents any term in the sequence using the variable \(k\) which indicates the position of the term.

For example, in the sequence given in the exercise, \[\frac{1}{e} + \frac{2}{e^{2}} + \frac{3}{e^{3}} + \cdots + \frac{n}{e^{n}}\], each term can be represented as \[\frac{k}{e^{k}}\], where \(k\) is an integer starting from 1 to \(n\).

Hence, the formula \[\frac{k}{e^{k}}\] serves as the general term for this sequence. Identifying the general term is a pivotal step as it simplifies the representation of sequences and series, making further calculations more manageable.

summation notation

Summation notation, also known as sigma notation, is a compact way to represent the sum of a series. The Greek letter \( \Sigma \) (sigma) is used to express the sum of terms in a sequence.

The notation \(\textstyle\sum_{k=1}^{n} a_k\) indicates that you are summing the terms \(a_k\) starting from \(k=1\) up to \(k=n\).

In the exercise example, the series \(\frac{1}{e} + \frac{2}{e^{2}} + \frac{3}{e^{3}} + \cdots + \frac{n}{e^{n}}\) can be written using summation notation as \(\sum_{k=1}^{n} \frac{k}{e^{k}}\).

This notation makes it simpler to handle complex series and perform operations on them. By learning summation notation, you can easily visualize and compute the sums associated with sequences.