Mathematical Tools for Physics

2—Infinite Series 58

2.14 Use series expansions to evaluate

lim

x→ 0

1 −cosx

1 −coshx

and lim

x→ 0

sinkx

x

2.15 Evaluate using series

lim

x→ 0

(

1

sin^2 x

−

1

x^2

)

Now do it again, setting up the algebra differently and finding an easier (or harder) way. Ans: 1 / 3

2.16 For some more practice with series, evaluate

lim

x→ 0

(

2

x

+

1

1 −

√

1 +x

)

Ans: Check experimentally with a pocket calculator.

2.17 Expand the integrand and find the power series expansion for

ln(1 +x) =

∫x

0

dt

1 +t

2.18 The error functionerf(x)is defined by an integral. Expand the integrand, integrate term by term, and
develop a power series representation forerf. For what values ofxdoes it converge? Evaluateerf(1)from this
series and compare it to the result of problem1.34. Also, as further validation of the integral in problem1.13,
do the power series expansion of both sides and verify the expansions of the two sides of the equation.

2.19 Verify that the combinatorial factormCnis really what results for the coefficients when you specialize the
binomial series Eq. ( 3 ) to the case that the exponent is an integer.