Mathematical Tools for Physics

2—Infinite Series 64

Iterate on this and compare it to the series expansion of the exact solution.
Solve 0. 001 x^2 +x+ 1 = 0.

2.41 Evaluate the limits

lim

x→ 0

sinx−tanx

x

, lim

x→ 0

sinx−tanx

x^2

, lim

x→ 0

sinx−tanx

x^3

,

2.42 Fill in the missing steps in the derivation of Eq. ( 19 ).

2.43 Is the result in Eq. ( 19 ) normalized properly? What is its integraldδover allδ? Ans: 1

2.44 A political survey asks 1500 people randomly selected from the entire country whom they will vote for as
dog-catcher-in-chief. The results are 49.0% for I. Hulk and 51.0% for Spiderman. Assuming these numbers are
representative, what is the probability that Mr. Hulk will win the final vote? What would the answer be if the
survey had asked 150 or 15000 people with the same 49-51 results? Ans:22% = 0. 5

(

1 −erf(15/ 27 .38)

)

2.45 For the function defined in problem 38 , what is its behavior nearx= 1? Compare this result to equation

(1.4). Note: the integral is

∫Λ

0 +

∫x

Λ. Also,^1 −t

(^2) = (1 +t)(1−t), and this≈2(1−t)near 1.
2.46 What is the expansion in powers of 1 /(1 +t^2 )for smallt. That was easy, now what is it for larget? In
each case, what is the domain of convergence?
2.47 The “average” of two numbersaandbcommonly means(a+b)/ 2 , the arithmetic mean. There are many
other averages however. (a,b > 0 )
Mn(a,b) =

[

(an+bn)/ 2

] 1 /n

is thenthmean. Show that this includes the geometric mean as a special case:

√

ab= limn→ 0 Mn(a,b).

2.48 Using the definition from the preceding problem, show thatdMn/dn > 0. From this, write down a sequence
of inequalities for various means: arithmetic, geometric, rms, harmonic.