Solved: "Use the formula for the sum of a..."

sodni3

Answered question

2021-01-31

Use the formula for the sum of a geometric series to find the sum, or state that the series diverges.

\sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}}

brawnyN

Skilled2021-02-01Added 91 answers

Given:

\sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}}

To find:
Using the sum geometric series to find the sum of given series.
Consider,

\sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}}

It can be written as,

\sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}} = \sum_{n = 0}^{\infty} [\frac{3 (- 2)^{n}}{8^{n}} - \frac{5^{n}}{8^{n}}]

Apply the sum rule:

\sum a_{n} + b_{n} = \sum a_{n} + \sum b_{n}

= \sum_{n = 0}^{\infty} \frac{3 (- 2)^{n}}{8^{n}} - \sum_{n = 0}^{\infty} \frac{5^{n}}{8^{n}}

First to find the value of

\sum_{n = 0}^{\infty} \frac{3 (- 2)^{n}}{8^{n}}

\Rightarrow \sum_{n = 0}^{\infty} \frac{3 (- 2)^{n}}{8^{n}} = 3 \sum_{n = 0}^{\infty} \frac{(- 2)^{n}}{8^{n}}

Applying the property:

\frac{a^{n}}{b^{n}} = (\frac{a}{b})^{n}

= 3 \sum_{n = 0}^{\infty} (\frac{- 2}{8})^{n}

Using the geometric series:

\sum_{n = 0}^{\infty} (\frac{- 2}{8})^{n} = \frac{1}{1 - (- \frac{2}{8})}

= \frac{1}{(1 + \frac{2}{8})}

= \frac{1}{(\frac{8 + 2}{8})}

= \frac{1}{(\frac{10}{8})}

= \frac{8}{10}

\sum_{n = 0}^{\infty} (\frac{- 2}{8})^{n} = \frac{4}{5}

\Rightarrow \sum_{n = 0}^{\infty} \frac{3 (- 2)^{n}}{8^{n}} = 3 \times \frac{4}{5} = \frac{12}{5}

Now to find the value of

\Rightarrow \sum_{n = 0}^{\infty} \frac{5^{n}}{8^{n}}

\Rightarrow \sum_{n = 0}^{\infty} \frac{5^{n}}{8^{n}} = (\frac{5}{8})^{n}

= \frac{1}{(1 - \frac{5}{8})}

= \frac{1}{(\frac{8 - 5}{8})}

= \frac{1}{(\frac{3}{8})}

= \frac{8}{3}

\sum_{n = 0}^{\infty} \frac{5^{n}}{8^{n}} = \frac{8}{3}

\Rightarrow \sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}} = \frac{12}{5} - \frac{8}{3}

= \frac{36 - 40}{15}

= \frac{- 4}{15}

\Rightarrow \sum_{n = 0}^{\infty} \frac{3 (- 2)^{n} - 5^{n}}{8^{n}} = - \frac{4}{15}

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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