MATHEMATICAL ANALYSIS PROBLEMS LIST 12
18.12.08
(1) Give the formula for Cn = Xn
i=1
b − a n f¡
a + ib − a n
¢, and the
compute lim
n→∞Cn
(a) f(x) = 1, a = 5, b = 8; (b) f(x) = x, a = 0, b = 1, (c) f(x) = x, a = 1, b = 5; (d) f(x) = x2, a = 0, b = 5, (e) f(x) = x3, a = 0, b = 1; (f) f(x) = 2x+5, a = −3, b = 4, (g) f(x) = x2+ 1, a = −1, b = 2,
(h) f(x) = x3+x, a = 0, b = 4; (i) f(x) = ex, a = 0, b = 1. (2) Compute the following denite integrals by constructing a sequ- ence of partitions of the interval, corresponding Riemann sums, and their limit:
(a) Z 4
2
x10dx, (ti = 2 · 2i/n), (b) Z e
1
log x
x dx, (ti = ei/n), (c)
Z 20
0
x dx, (d) Z 10
1
e2xdx, (e)
Z 1
0
√3
x dx, (ti = ni33), (f) Z 1
−1
|x| dx, (g)
Z 2
1
dx
x dx, (ti = 2i/n), (h) Z 4
0
√x dx, (ti = 4in22).
(3) Compute the denite integrals:
(a) Z π
−π
sin x2007dx, (b) Z 2
0
arctan([x]) dx, (c)
Z 2
0
[cos(x2)] dx, (d) Z 1
0
√1 + x dx,
(e) Z −1
−2
1
(11 + 5x)3 dx, (f) Z 2
−13
1 p5
(3 − x)4 dx, (g)
Z 1
0
x
(x2+ 1)2 dx, (h) Z 3
0 sgn (x3− x) dx, (i)
Z 1
0
x e−xdx, (j) Z π/2
0
x cos x dx,
1
(k) Z e−1
0
log(x + 1) dx, (l) Z π
0
x3 sin x dx, (m)
Z 9
4
√x
√x − 1dx, (n) Z e3
1
1 x√
1 + log xdx, (o)
Z 2
1
1
x + x3 dx, (p) Z 2
0
√ 1
x + 1 +p
(x + 1)3dx, (q)
Z 5
0
|x2− 5x + 6| dx, (r) Z 1
0
ex
ex− e−x dx, (s)
Z 2
1
x log2x dx, (t) Z √7
0
x3
√3
1 + x2 dx, (u)
Z 6π
0
| sin x| dx, (w) Z π/2
0
cos x sin11x dx, (x)
Z log 5
0
ex√ ex− 1
ex+ 5 dx, (y) Z π
−π
x2007cos x dx, (z)
Z 2π
0
(x − π)2007cos x dx. (4) Prove the following estimates:
(a) Z π/2
0
sin x
x dx < 2, (b) 1 5 <
Z 2
1
1
x2+ 1dx < 1 2, (c) 1
11 <
Z 10
9
1
x + sin xdx < 1
8, (d) Z 2
−1
|x|
x2+ 1dx < 3 2, (e)
Z 1
0
x(1 − x99+x) dx < 1
2, (f) 2√ 2 <
Z 4
2
x1/xdx, (g) 5 <
Z 3
1
xxdx < 31, (h) Z 2
1
1
xdx < 3 4. (5) Compute the following limits:
(a) limn→∞n1 +n+11 +n+21 +n+31 + · · · + 2n1 , (b) lim
n→∞
120+220+320+···+n20 n21 , (c) limn→∞³
1
n2 +(n+1)1 2 +(n+1)1 2 + (n+3)1 2 + · · · + (2n)1 2
´
· n, (d) lim
n→∞
√ 1 n√
2n +√n√12n+1 + √n√12n+2 +√n√12n+3 + · · · + √n1√3n, (e) lim
n→∞
¡sinn1 + sin2n+ sinn3 + · · · + sinnn¢
·n1, (f) limn→∞¡√
4n +√
4n + 1 +√
4n + 2 + · · · +√ 5n¢
· n√1n, (g) lim
n→∞
³ 1
√3
n+ √3 1
n+1 + √3 1
n+2+ · · · + √31
8n
´
· √31
n2, (h) limn→∞ √6n·(√√n+3n+√n+1+√3n+1+√n+2+···+√3n+2+···+√2n√32n),
(i) lim
n→∞
n
n2 + n2n+1 + n2n+4 +n2n+9 + n2n+16+ · · · + n2+nn 2, (j) lim
n→∞
4
5n +5n+34 + 5n+64 +5n+94 + · · · + 26n4 ,
2
(k) limn→∞7n1 +7n+21 + 7n+41 +7n+61 + · · · + 9n1 , (l) lim
n→∞
1
7n2 + 7n21+1 + 7n21+2 +7n21+3 + · · · + 8n12, (m) limn→∞n1 ³
e√1
n + e√2
n + e√3
n + · · · + e√n
n
´,
(n) limn→∞³
√1
n+√n+31 +√n+61 + √n+91 + · · · + √17n
´ √1 n, (o) lim
n→∞
n2+0
(3n)3 + (3n+1)n2+13 +(3n+2)n2+23 + (3n+3)n2+33 + · · · + n(4n)2+n3, (p) lim
n→∞
n
2n2 + 2(n+1)n 2 +2(n+2)n 2 +2(n+3)n 2 + · · · + 50nn2, (r) limn→∞2nn2 + n2+(n+1)n 2 + n2+(n+2)n 2 +n2+(n+3)n 2 + · · · + 50nn2.
3