Question

Find the sum for each of the series: a. \(5+\frac{25}{7}+\frac{125}{49}+\frac{625}{343}+\cdots\) b. \(\sum_{n=0}^{\infty} \frac{(-1)^{n} 3}{4^{n}}\) c. \(\sum_{n=2}^{\infty} \frac{2}{5^{n}}\). d. \(\sum_{n=-1}^{\infty}(-1)^{n+1}\left(\frac{e}{\pi}\right)^{n}\). e. \(\sum_{n=0}^{\infty}\left(\frac{5}{2^{n}}+\frac{1}{3^{n}}\right)\). f. \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\) g. What is \(0.569 ?\)

Short answer

a. 35, b. \(\frac{4}{5}\), c. \(\frac{2}{15}\), d. \(\frac{e}{\pi + e}\), e. \(13\frac{1}{2}\), f. \(\frac{11}{6}\), g. 0.569

Step-by-step solution

Solve part (a)

Identify the first term \(a = 5\), and the common ratio \(r = \frac{25}{7} \div 5 = \frac{5}{7}\). Using the formula for the sum of an infinite geometric series gives \[S = \frac{5}{1 - \frac{5}{7}} = 35\]

Solve part (b)

Identify the first term \(a = 3\) and the common ratio \(r = \frac{(-1) 3}{4}\div 3 = -\frac{1}{4}\). The sum becomes \[S = \frac{3}{1 - -\frac{1}{4}} = \frac{4}{5}\]

Solve part (c)

The first term is \(a = \frac{2}{5^2} = \frac{2}{25}\) and the common ratio is \(r = \frac{2}{5}\). The sum is \[S = \frac{\frac{2}{25}}{1 - \frac{2}{5}} = \frac{2}{15}\]

Solve part (d)

The first term is \(a = \frac{e}{\pi}\) and the common ratio is \(r = -\frac{e}{\pi}\). So, \[S = \frac{\frac{e}{\pi}}{1 - -\frac{e}{\pi}} = \frac{e}{\pi + e}\]

Solve part (e)

This is a sum of two geometric series. The first has \(a_1 = 5\) and \(r_1 = \frac{1}{2}\), and the second has \(a_2 = 1\) and \(r_2 = \frac{1}{3}\). The sum of the two series is \[S= \frac{5}{1 - \frac{1}{2}} + \frac{1}{1 - \frac{1}{3}} = 10 + \frac{3}{2} = 13\frac{1}{2}\]

Solve part (f)

First, split the given fraction into partial fractions \[\frac{3}{n(n+3)} = \frac{1}{n} - \frac{1}{n+3}\] Then add up the first few terms to notice the sequence \[1 - \left(\frac{1}{4} + \frac{1}{4} - \left(\frac{1}{7} + \frac{1}{7}\right)\right)\] A pattern emerges and the sum becomes \[S = 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6}\]

Solve part (g)

This is not a series, but a decimal, therefore there is no need for computation. Thus, \(0.569 = 0.569\)

Key concepts

Geometric Series

A geometric series is a mathematical series where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the "common ratio." This type of series can be finite or infinite and has a specific formula that allows for quick calculation of its sum.

For an infinite geometric series, the formula to find the sum is given by:\[S = \frac{a}{1 - r}\]where:

• \(a\) is the first term of the series,
• \(r\) is the common ratio, and
• \(-1 < r < 1\), to ensure convergence of the series.

This formula is especially useful because it allows us to find the total sum of an infinite series quickly, without having to manually add each term.

Mathematical Series

Mathematical series are sums of sequences of numbers. They can be arithmetic, geometric, or more complex, like harmonic or logarithmic series. Series are vital in mathematical analysis and calculus, helping us understand convergence and divergence.

A series \(\sum_{n=0}^{\infty} a_n\) represents the sum of an infinite number of terms \(a_n\). The behavior of a series—whether it diverges to infinity or converges to a specific value—depends heavily on the properties of \(a_n\).

Mathematical series are used in various applications such as calculating compound interest, analyzing algorithms, and even in signal processing. Understanding series is fundamental for anyone studying higher mathematics.

Partial Fractions

Partial fractions are used to break down complex rational expressions into simpler fractions that are easier to integrate or sum. This technique is particularly useful when dealing with rational functions in calculus.

For the series \(\sum_{n=1}^{\infty} \frac{3}{n(n+3)}\), we use partial fractions to rewrite the expression:\[\frac{3}{n(n+3)} = \frac{1}{n} - \frac{1}{n+3}\]By splitting the original fraction, we can simplify the sum of the series.

This simplification often reveals a telescoping pattern, making it easier to find the sum. Understanding partial fractions is essential for tackling complex series and integrals.

Common Ratio

In the context of geometric series, the common ratio is the constant factor between consecutive terms. It is a crucial element in determining the rate at which the terms of the series decrease or increase.

To find the common ratio \(r\), we divide any term by its preceding term. For instance, in the series \(5, \frac{25}{7}, \frac{125}{49}\), the common ratio is:\[r = \frac{\frac{25}{7}}{5} = \frac{5}{7}\]The common ratio provides insight into the behavior of the series, especially in infinite series.
If \(-1 < r < 1\), the series converges; otherwise, it diverges. Understanding the common ratio helps in determining whether a series has a finite sum or not.