# x, a = 0, b = 1, (c) f(x

## Full text

(1)

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 denite 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 denite 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

(2)

(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

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

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 +n12n+1 + n12n+2 +n12n+3 + · · · + n13n, (e) lim

n→∞

¡sinn1 + sin2n+ sinn3 + · · · + sinnn¢

·n1, (f) limn→∞¡√

4n +√

4n + 1 +√

4n + 2 + · · · +√ 5n¢

· n1n, (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+···+2n32n),

(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

(3)

(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.

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