Consider the following series.1+11248.10Determine whether the series is convergent or divergent. Justify Four answer.O Converges; the series is a constant multiple geometric series.Converges; the limit of the terms, a, is 0 as n goes to infinity.O Diverges; the limit of the terms, an, is not 0 as n goes to infinity.O Diverges; the series is a constant multiple of the harmonic series.If it is convergent, find the sum. (If the quantity diverges, enter DIVERGES.)

Given that: The series is 12 + 14 +16 +18+110 + ... .

Answered: O Converges; the series is a constant…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 81E

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Question

Consider the following series.
1
+
1
1
2
4
8.
10
Determine whether the series is convergent or divergent. Justify Four answer.
O Converges; the series is a constant multiple geometric series.
Converges; the limit of the terms, a, is 0 as n goes to infinity.
O Diverges; the limit of the terms, an, is not 0 as n goes to infinity.
O Diverges; the series is a constant multiple of the harmonic series.
If it is convergent, find the sum. (If the quantity diverges, enter DIVERGES.)

Expert Solution

Step 1

Given that:

The series is $\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + . . .$ .

Step 2

By using,

The sum of the infinite geometric series is give by a + a r + ar² + . . . + a r^{n - 1} + . . . = $\sum_{n = 0}^{\infty} a r^{k}$ ,

where a is the first term and r is the ratio.

The sum converges to $\frac{a}{1 - r} i f |r| < 1$ and diverges if $|r| \geq 1$ .

Step 3

Consider the sum,

$\frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + . . . . = \frac{1}{2} [1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + . . . .] = \frac{1}{2} \sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n} \Rightarrow \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} + . . . . = \frac{1}{2} \sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n}$

$\sum_{n = 0}^{\infty} {(\frac{1}{2})}^{n}$ is the geometric series with first term a = 1 and the common ratio ( r ) = $\frac{1}{2}$ .

Also , r = $|\frac{1}{2}|$ < 1

Then,

It is convergent .

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