Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

Summary of methods for evaluating limits

To find the limit of a continuous functionf(x) at a pointx=a, simply substitute

the valueainto the function noting that∞^0 = 0 and that∞ 0 =∞.Theonly

difficulty occurs when either of the expressions^00 or ∞∞results. In this case

differentiate top and bottom and try again. Continue differentiating until the top

and bottom limits are no longer both zero or both infinity. If the undetermined

form 0×∞occurs then it can always be rewritten as^00 or∞∞.

4.8 Exercises

4.1 Sum the even numbers between 1000 and 2000 inclusive.

4.2 If you invest£1000 on the first day of each year, and interest is paid at 5% on
your balance at the end of each year, how much money do you have after 25
years?

4.3 How does the convergence of the series

∑∞

n=r

(n−r)!

n!

depend on the integerr?

4.4 Show that for testing the convergence of the series

x+y+x^2 +y^2 +x^3 +y^3 +···,

where 0<x<y<1, the D’Alembert ratio test fails but the Cauchy root test is

successful.

4.5 Find the sumSNof the firstNterms of the following series, and hence determine
whether the series are convergent, divergent or oscillatory:

(a)

∑∞

n=1

ln

(

n+1

n

)

, (b)

∑∞

n=0

(−2)n, (c)

∑∞

n=1

(−1)n+1n

3 n

.

4.6 By grouping and rearranging terms of the absolutely convergent series

S=

∑∞

n=1

1

n^2

,

show that

So=

∑∞

nodd

1

n^2

=

3 S

4

.

4.7 Use the difference method to sum the series

∑N

n=2

2 n− 1

2 n^2 (n−1)^2

.