   Chapter 4.4, Problem 79E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# If an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is A = A 0 ( 1 + r n ) n t If we let n →∞, we refer to the continuous compounding of interest. Use 1’Hospital’s Rule to show that if interest is compounded continuously, then the amount after t years is A = A 0 e r t

To determine

To prove: An amount A0 which is compounded continuously for t years is, A=A0ert

Explanation

Given:

An amount A0 is invested at a compound interest with rate r compounded n times a year.

The amount after t years is given by, A=A0(1+rn)nt .

Proof:

The required amount is calculated by, limnA=limnA0(1+rn)nt . (1)

As A0 is constant, consider the following limit.

Let, y=limn(1+rn)nt . (2)

Take natural logarithm on both sides,

lny=ln(limn(1+rn)nt)=limx(ln(1+rn)nt)=limxnt(ln(1+rn))=limxln(1+rn)1nt

Therefore, lny=limxln(1+rn)1nt . (3)

Obtain the value of the function as n approaches .

As n approaches , the numerator is,

ln(1+rn)=ln(1+r)=ln(1+0)=ln(1)=0

As n approaches , the denominator is,

1nt=1=0

Thus, limxln(1+rn)1nt=00 is in an indeterminate form

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