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Chapter 24: Problem 6

For the series \(5-\frac{5}{2}+\frac{5}{4}-\frac{5}{8}+\cdots+\frac{(-1)^{n}-5}{2^{n-1}}\) find an expression for the sum of the first \(n\) terms. Also if the series converges, find the sum to infinity.

Short Answer

Expert verified
The sum of the first n terms of the series is \(S_n = \frac{10(1-(-\frac{1}{2})^n)}{3}\), and the sum to infinity is \(S_{\infty} = \frac{10}{3}\).

Step by step solution

01

Identify the general term#a

For the given series \(5-\frac{5}{2}+\frac{5}{4}-\frac{5}{8}+\cdots+\frac{(-1)^{n}-5}{2^{n-1}}\), we observe that the general term can be described as: \[a_{n} = \frac{(-1)^{n}-5}{2^{n-1}}\]
02

Test for convergence#a

To test if the series is convergent, we consider the ratio of consecutive terms, denoted by r: \[r = \frac{a_{n+1}}{a_{n}} = \frac{(-1)^{n+1}-5}{(-1)^n-5} \cdot \frac{2^{n-1}}{2^n}\] Simplifying this expression, we get: \[ r = \frac{(-1)^{n+1}-5}{2(-1)^n+10} \] Since the absolute value of r is less than 1, i.e., \(|r| = \frac{1}{2}\) (the ratio between consecutive terms), the series converges.
03

Find the sum of the first n terms#a

As we have established that the series converges and can be represented as a geometric series, we can find the sum of the first n terms, denoted by S_n: \[S_n = \frac{a_1(1-r^n)}{1-r}\] We have \(a_1 = 5\) and \(r = -\frac{1}{2}\). Substitute these values into the formula: \[ S_n = \frac{5(1-(-\frac{1}{2})^n)}{1-(-\frac{1}{2})} \] Simplify the expression: \[ S_n = \frac{5(1-(-\frac{1}{2})^n)}{\frac{3}{2}} = \frac{10(1-(-\frac{1}{2})^n)}{3} \] So, the sum of the first n terms of the series is: \[S_n = \frac{10(1-(-\frac{1}{2})^n)}{3}\]
04

Find the sum to infinity, if it exists#a

Since the given series converges, we can find the sum to infinity using the formula: \[S_{\infty} = \frac{a_1}{1-r}\] With \(a_1 = 5\) and \(r = -\frac{1}{2}\), we insert the values into the formula: \[ S_{\infty} = \frac{5}{1-(-\frac{1}{2})} = \frac{5}{\frac{3}{2}} \] Which simplifies to: \[S_{\infty} = \frac{10}{3}\] Thus, the sum to infinity of the given series is \(\frac{10}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of n Terms
Understanding the sum of the first n terms in a sequence is crucial for solving many problems in mathematics. When dealing with a series, particularly geometric ones, we have a formula that governs the sum of its first n terms:

For a geometric series with a common ratio r, and the first term a_1, the sum S_n for the first n terms can be calculated using the formula:
\[S_n = \frac{a_1(1-r^n)}{1-r}\]
In our exercise example, the first term is 5 and the common ratio is \(-\frac{1}{2}\). By substituting these values into the formula, we find a simplified expression to calculate the sum of the first n terms. It's important to note that this formula works specifically for geometric series where the terms are produced by multiplying the previous term by a constant ratio.
Geometric Series
A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (denoted as r).

Characteristic Features

The series has a distinct pattern and can be represented as a, ar, ar^2, ar^3, ..., where a is the first term. Geometric series can converge or diverge depending on the value of r. If \(|r| < 1\), the series converges. In contrast, if \(|r| \geq 1\), the series will diverge. Convergence of a geometric series leads to a finite sum, even as the number of terms grows infinitely large鈥攚hich brings us to the concept of the sum to infinity.
Sum to Infinity
When a geometric series converges, we can calculate the sum to infinity, denoted as 厂冲鈭�. This is the limit of the sum of the first n terms as n approaches infinity. The formula to calculate the sum to infinity for a geometric series is:
\[S_{\infty} = \frac{a_1}{1-r}\]
In the context of our example, where the common ratio r is \(-\frac{1}{2}\) and the first term a_1 is 5, the sum to infinity is calculated using the given formula. We arrive at a finite number, \(\frac{10}{3}\), signifying the sum of an infinite number of terms in our specific geometric series.
Convergence Test
To determine whether a given geometric series will converge to a finite sum or not, we apply a convergence test. Generally, for a geometric series with a common ratio r, the series will converge if the absolute value of the common ratio is less than 1, that is, \(|r| < 1\).

Application of the Convergence Test

In our exercise, the ratio between consecutive terms is \(|r| = \frac{1}{2}\), which satisfies the condition for convergence. Thus, we conclude that the series is convergent. It鈥檚 this property of convergence that allows us to find sums to infinity and apply formulas which only hold for series that do not diverge to infinity.

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Most popular questions from this chapter

Find the range of values of \(x\) for which the following series is convergent: \(\frac{(x-2)}{1}+\frac{(x-2)^{2}}{2}+\frac{(x-2)^{3}}{3}+\ldots+\frac{(x-2)^{n}}{n}+\ldots\)

Investigate the convergence of the series $$ \frac{1}{1 \times 2}+\frac{x}{2 \times 3}+\frac{x^{2}}{3 \times 4}+\frac{x^{3}}{4 \times 5}+\ldots \text { for } x>0 $$

Take your time and work carefully. Determine whether each of the following series is convergent: (a) \(\frac{2}{2 \times 3}+\frac{2}{3 \times 4}+\frac{2}{4 \times 5}+\frac{2}{5 \times 6}+\ldots\) (b) \(\frac{2}{1^{2}}+\frac{2^{2}}{2^{2}}+\frac{2^{3}}{3^{2}}+\frac{2^{4}}{4^{2}}+\ldots+\frac{2^{n}}{n^{2}}+\ldots\) (c) \(u_{n}=\frac{1+2 n^{2}}{1+n^{2}}\) (d) \(u_{n}=\frac{1}{n !}\)

Find the sum of \(n\) terms of the series \(1^{2}+3^{2}+5^{2}+\ldots+(2 n-1)^{2}\)

Show that the series \(1+\frac{2 x}{5}+\frac{3 x^{2}}{25}+\frac{4 x^{3}}{125}+\ldots\) is convergent if \(-5
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