Advanced book on Mathematics Olympiad

3.2 Continuity, Derivatives, and Integrals 153

465.Letn≥0 be an integer. Compute the integral
∫π

0

1 −cosnx

1 −cosx

dx.

466.Compute the integral

In=

∫ π 2

0

sinnxdx.

Use the answer to prove the Wallis formula

lim

n→∞

[

2 · 4 · 6 ··· 2 n

1 · 3 · 5 ···( 2 n− 1 )

] 2

·

1

n

=π.

467.Compute
∫π

−π

sinnx

( 1 + 2 x)sinx

dx, n≥ 0.

3.2.9 Riemann Sums

The definite integral of a function is the area under the graph of the function. In ap-
proximating the area under the graph by a family of rectangles, the sum of the areas of
the rectangles, called a Riemann sum, approximates the integral. When these rectangles
have equal width, the approximation of the integral by Riemann sums reads

lim

n→∞

1

n

∑n

i= 1

f(ξi)=

∫b

a

f(x)dx,

where eachξiis a number in the interval[a+i−n^1 (b−a), a+ni(b−a)].
Since the Riemann sum depends on the positive integern, it can be thought of as the
term of a sequence. Sometimes the terms of a sequence can be recognized as the Riemann
sums of a function, and this can prove helpful for finding the limit of the sequence. Let us
show how this works, following Hilbert’s advice: “always start with an easy example.’’

Example.Compute the limit

lim

n→∞

(

1

n+ 1

+

1

n+ 2

+···+

1

2 n

)

.

Solution.If we rewrite

1

n+ 1

+

1

n+ 2

+···+

1

2 n