f(x) = sin x, a = /2

In the bouncing ball example above, find the height of the tenth rebound, and the distance traveled by the ball after it touches the ground the tenth time. Compare this distance with the total distance traveled.

Derive the formula (1.4) for the sum Sn of the geometric progression Sn = a + ar + ar2 + + arn1. Hint: Multiply Sn by r and subtract the result from Sn; then solve for Sn. Show that the geometric series (1.6) converges if and only if |r| < 1; also show that if |r| < 1, the sum is given by equation (1.8).

0.55555

0.818181

0.583333

0.61111

0.185185

0.694444

0.857142857142

0.576923076923076923

0.678571428571428571

In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n = 2, the water can be made as pure as you like, but that if n = 3, at least one-half of the impurity will remain no matter how many stages are used.

If you invest a dollar at 6% interest compounded monthly, it amounts to (1.005)n dollars after n months. If you invest $10 at the beginning of each month for 10 years (120 months), how much will you have at the end of the 10 years?

A computer program gives the result 1/6 for the sum of the series P n=0(5)n. Show that this series is divergent. Do you see what happened? Warning hint: Always consider whether an answer is reasonable, whether its a computer answer or your work by hand.

Connect the midpoints of the sides of an equilateral triangle to form 4 smaller equilateral triangles. Leave the middle small triangle blank, but for each of the other 3 small triangles, draw lines connecting the midpoints of the sides to create 4 tiny triangles. Again leave each middle tiny triangle blank and draw the lines to divide the others into 4 parts. Find the infinite series for the total area left blank if this process is continued indefinitely. (Suggestion: Let the area of the original triangle be 1; then the area of the first blank triangle is 1/4.) Sum the series to find the total area left blank. Is the answer what you expect? Hint: What is the area of a straight line? (Comment: You have constructed a fractal called the Sierpinski gasket. A fractal has the property that a magnified view of a small part of it looks very much like the original.)

Suppose a large number of particles are bouncing back and forth between x = 0 and x = 1, except that at each endpoint some escape. Let r be the fraction reflected each time; then (1 r) is the fraction escaping. Suppose the particles start at x = 0 heading toward x = 1; eventually all particles will escape. Write an infinite series for the fraction which escape at x = 1 and similarly for the fraction which escape at x = 0. Sum both the series. What is the largest fraction of the particles which can escape at x = 0? (Remember that r must be between 0 and 1.)

Use the integral test to show that P n=0 en2 converges. Hint: Although you cannot evaluate the integral, you can show that it is finite (which is all that is necessary) by comparing it with R endn.

X n=1 2n n2

n=0 3n 22n

X n=0 n! (2n)!

n=0 5n(n!)2 (2n)!

n=1 10n (n!)2

n=1 n! 100n

n=0 32n 23n

n=0 en n

n=0 (n!)3e3n (3n)!

n=0 100n n200

X n=0 n!(2n)! (3n)!

n=0 p(2n)! n!

Prove the ratio test. Hint: If |an+1/an| < 1, take so that << 1. Then |an+1/an| < if n is large, say n N. This means that we have |aN+1| < |aN |, |aN+2| < |aN+1| < 2|aN |, and so on. Compare with the geometric series X n=1 n|aN |. Also prove that a series with > 1 diverges. Hint: Take >> 1, and use the preliminary test.

X n=9 (2n + 1)(3n 5) n2 73

X n=0 n(n + 1) (n + 2)2(n + 3)

n=5 1 2n n2

n=1 n2 + 3n + 4 n4 + 7n3 + 6n 3

X n=3 (n ln n) 2 5n4 3n2 + 1

n=1 n3 + 5n 1 n2 sin n3

n=1 n3 + 5n 1 n2 sin n3

ecos x Hint: ecos x = e ecos x1 .

f(x) = sin x, a = /2

f(x) = 1 x, a = 1

f(x) = e x, a = 3

f(x) = cos x, a =

f(x) = cot x, a = /2

f(x) = x a = 25