Mathematical Tools for Physics

2—Infinite Series 59

2.20 Determine the double power series representation about(0,0)of

1

(1−x/a)(1−y/b)

2.21 Determine the double power series representation about(0,0)of

1

1 −x/a−y/b

2.22 Use a pocket calculator that can handle 100! and find the ratio of Stirling’s approximation to the exact
value. You may not be able to find the difference of two such large numbers. An improvement on the basic
Stirling’s formula is
√
2 πnnne−n

(

1 +

1

12 n

)

What is the ratio of approximate to exact forn= 1,2, 10?
Ans: 0. 99898 , 0. 99948 ,...

2.23 Evaluate the sum

∑∞

1 1 /n(n+ 1). To do this, write the single term^1 /n(n+ 1)as a combination of two
fractions with denominatornand(n+ 1)respectively, then start to write out the stated infinite series to a few
terms to see the pattern. When you do this you may be tempted to separate it into two series, of positive and of
negative terms. Examine the problem of convergence and show why this is wrong. Ans: 1

2.24 You can sometimes use the result of the previous problem to improve the convergence of a slow-converging
series. The sum

∑∞

1 1 /n

(^2) converges, but not very fast. If you add zero to it you don’t change the answer,
but if you’re clever about how you add it you can change this into a much faster converging series. Add
1 −

∑∞

1 1 /n(n+ 1)to this series and combine the sums.