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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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BuyFindarrow_forward

Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem

Determining Convergence or Divergence In Exercises 37–42, determine whether the geometric series converges or diverges. If it converges, find its sum.

n = 0 [ ( 0.5 ) n + ( 0.2 ) n ]

To determine

To calculate: The convergence or divergence of the geometric series n=0[(0.5)n+(0.2)n] if it converges then find its sum.

Explanation

Given Information:

The provided series is n=0[(0.5)n+(0.2)n].

Formula used:

The general form of Geometric series is n=0arn where r is the common ratio is and arn is the general term of the series.

If |r|1 then the series is said to be divergent.

If |r|<1 then the series converge to the sum n=0arn=a1r.

Calculation:

Consider the series n=0[(0.5)n+(0.2)n].

The geometric series can be rearranged as,

n=0[(0.5)n]+n=0[(0.2)n]

Compare provided series with the general geometric series n=0arn. From this,

For n=0[(0.5)n],

a=1

And

r=0.5

And,

For n=0[(0.2)n],

a=1

And

r=0.2

Apply the Geometric Series Test,

For n=0[(0.5)n]

In the expression n=0[(0

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