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

slaggingV 2021-02-06 Answered
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 remainder is less than 103.
c. Find lower and upper bounds (ln and Un, respectively) on the exact value of the series.
k=113k
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Neelam Wainwright
Answered 2021-02-07 Author has 102 answers

Given,
We are answering the first three subparts as per our honor code.
The series k=113k.Considering the series is convergent then we have to answer the following.
Calculation
(a). Find an upper bound for the remainder in terms of n.
Since we know that Rn<n13xdx
n13xdx=limbnb13xdx
=limb[1ln(3)3x]nb
=limb[1ln(3)3b+1ln(3)3n]
=0+1ln(3)3n
=1ln(3)3n
Hence an upper bound for the remainder in terms of n is =1ln(3)3n
(b) Find how many terms are needed to ensure that the remainder is less than 103.
Since given 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). Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series.
Since Sn+n+113xdx<S<Sn+n13xdx 

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

Not exactly what you’re looking for?
Ask My Question
Jeffrey Jordon
Answered 2021-12-16 Author has 2262 answers

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions