Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the

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Answered question

2021-02-06

a. Find an upper bound for the remainder in terms of n. 
b. Find how many terms are needed to ensure that the remainder is less than 103
c. Find lower and upper bounds (ln and Un, respectively) on the exact value of the series. 
k=113k

Answer & Explanation

Neelam Wainwright

Neelam Wainwright

Skilled2021-02-07Added 102 answers

(a). We know  Rn<n13xdx
n13xdx=limbnb13xdx
=limb[1ln(3)3x]nb
=limb[1ln(3)3b+1ln(3)3n]
=0+1ln(3)3n
=1ln(3)3n
An upper bound for the remainder in terms of n is =1ln(3)3n
(b) We have Rn<103
1ln(3)3n<1103
ln(3)>ln(1000)ln(ln(3))
3n>1000ln(3)
nln(3)>ln(1000)ln(ln(3))
n>3ln(ln(3))ln(3)
n>2.645
(c).  So Sn+n+113xdx<S<Sn+n13xdx 

Sn+1ln(3)3n+1<S<Sn+1ln(3)3n

Jeffrey Jordon

Jeffrey Jordon

Expert2021-12-16Added 2605 answers

Answer is given below (on video)

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