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Provide the first five terms of the sequence of partial sums for the given series.

j=0(1)j+1

Short Answer

Expert verified

{-1,1,-1,1,-1}.

Step by step solution

01

Step1. Given Information

Consider the geometric seriesj=0(1)j+1.

The objective is to provide the first five terms of partial sums for the given series.

The strategy to find the first five terms of partial sums for the given series is to find the first five terms of the seriesj=0(1)j+1.

02

Step2. First term

The first term of the seriesj=0(1)j+1is obtained by substitutingj=0in(1)j+1.

Therefore, the value atj=0is:

role="math" localid="1649270985224" (1)0+1=(1)1(Substituting)=-1

The first term of the seriesj=0(1)j+1is1.

03

Step3. Second term

The second term of the seriesj=0(1)j+1is obtained by substitutingj=1in(1)j+1.

Therefore, the value atj=1is:

(1)1+1=(1)2(Substituting)=1

The second term of the seriesj=0(1)j+1is1.

04

Step4. Third term

The third term of the seriesj=0(1)j+1is obtained by substitutingj=2in(1)j+1.

Therefore, the value atj=2is:

(1)2+1=(1)3(Substituting)=-1

The third term of the seriesj=0(1)j+1is1.

05

Step5. Fourth term

The fourth term of the seriesj=0(1)j+1is obtained by substitutingj=3in(1)j+1.

Therefore, the value atj=3is:

(1)3+1=(1)4(Substituting)=1The fourth term of the seriesj=0(1)j+1is1.

06

Step6. Fifth term

The fifth term of the seriesj=0(1)j+1is obtained by substitutingj=4in(1)j+1Therefore, the value atj=4is:(1)4+1=(1)5(Substituting)=-1The fifth term of the seriesj=0(1)j+1is1.

07

Step7. Partial sums

The first five terms in the sequence of partial sums are:

S1=1S2=S1+a2=1+1(Substitution)=0S3=S2+a3=0+(1)(Substitution)=1S4=S3+a4=1+1(Substitution)=0S5=S4+a5=0+(1)(Substitution)=-1

role="math" localid="1649271664888" Therefore, first five terms of partial sums for the given series is{1,1,1,1,1}.

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