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Chapter 1: Q14P (page 1)

A computer program gives the result16 for the sum of the series 鈭�n=0鈭�(-5)n. Show that this series is divergent. Do you see what happened? Warning hint: Always consider whether an answer is reasonable, whether it鈥檚 a computer answer or your work by hand.

Short Answer

Expert verified

The infinite series鈭�n=0鈭�-5n is divergent.

Step by step solution

01

Explanation of Solution

Consider Geometric Series 鈭�n=0鈭�-5n.

02

Geometric series

A geometric series is a collection of numbers with a common ratio. This means that each phrase is found by multiplying the term before it.

The formula is provided by,

G.P=arn-1r-1

Here,

a is the first term of series,

r is the common ratio,

n is the total time period.

03

Calculation

The series is considered divergent since the total does not evaluate to any given value.

鈭�n=0鈭�-5n=-50+-51+-52+-53+-54+-55+....=1+-5+25+-125+625+-3125+....=?

The infinite summation formula was used by the computer software, which was a mistake.

鈭�n=0鈭�crn=c1-r,

Which only applies if r<1.

鈭�n=0鈭�-5n鈮�11--5=16
Hence, the required result is shown.

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Most popular questions from this chapter

Find a two-term approximation for each of the following integrals and an error bound for the given t interval.鈭�0te-x2dx,0<t<0.1

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Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point.

鈭�n=0鈭�(-1)n(2n)!(蟺2)2n

In(2-e-x)
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