Show that ∞ X n=1 (−1)n−1an is convergent

FINAL SAMPLE TEST

(1) Test the convergence/ divergence of the following series:

(a) X

n=1

 1 2 + 1

n− 1 n²

n

.

(b)

∞

X

n=1

(−1)ⁿ (2n + 1)!. (2) Let an=

Z ¹

n

0

e^t2dt for n ≥ 1. Show that

∞

X

n=1

(−1)ⁿ⁻¹an is convergent.

(3) Let sn= Z 1

0

xⁿ 1 +√

xdx. Compute lim

n→∞sn. (4) Compute the following definite integral

(a) Z π/2

0

sin³x cos⁴xdx (b)

Z 2π 0

x²cos 5xdx.

(c) Z ₅

−5

sin²⁰¹³x 1 + x²+ x⁴dx.

(d) Z π/2

0

cos x

sin²x + sin x − 2dx (e)

Z π/4 0

tan⁴xdx.

(5) Solve the following first order ordinary differential equation dy

dx − 3x²y = e^x3. (6) Find the area of the region between the graphs of

f (x) = 3x³− x²− 10x, g(x) = −x²+ 2x.

(7) Find the volume of the solid formed by revolving the region bounded by the graphs of

f (x) =√

x, g(x) = x² about the x-axis.

(8) Find the volume of the solid formed by revolving the region bounded by the graphs of y = x³+ x + 1, y = 1 and x = 1 about the line x = 2.

(9) Find the arc length of the graph of y = x³

6 + 1 2x on the interval [1/2, 2].

(10) Find the area of the surface formed by revolving the graph of f (x) = x³ on the interval [0, 1] about the x-axis.

1

2 FINAL SAMPLE TEST

(11) Show that

−

√2π

√1 + 2π ≤ Z 2π

0

sin x 1 + xdx ≤

√2π

√1 + 2π.