Calc 2

Worksheet #8 MAT 137 Sequences and Series

Name

Full credit will be given only where all reasoning and work is provided. Where appropriate, please enclose your final answers in boxes.

1. The Sierpinski Triangle is a fractal that can be constructed as follows: Start with a equilateral triangle and remove an equilateral triangle from the middle. On the next iteration, from each remaining triangle remove a central equilateral triangle as shown below, and continue forever.

Assume that the original area of the triangle is A. Construct a sequence that will tell how much area remains after the nth iteration and use this sequence to show that if this process is carried out an infinite number of times the remaining area is 0.

2. (a) Determine the limit of the sequence

{ 1

k −

1

k + 2

} k≥1

(b) Determine the sum of the series

∞∑ k=1

( 1

k −

1

k + 2

) 3. After graduating college you are offered a job with starting salary $50,000 per year. Your contract specifies a guaranteed

raise of 4% per year. Over a 40 year career in this job, how much have you made? Note: This is not an infinite series.

4. For what values of z will the geometric series ∑∞

n=0

(z + 3)n

2n converge and in terms of z what will it converge to?

5. Series of the form ∑∞

n=0

cn

n! are important in mathematics, physics, engineering, computer science,...

(a) Using the ratio test, show ∑∞

n=0

1

n! converges.

(b) Using the ratio test, show that ∑∞

n=0

cn

n! is convergent for ANY constant c ∈ R.

(c) Determine the partial sum s10 = ∑9

n=0

2n

n! (Write it out but use your calculator.)

(d) Compare your previous answer for s10 to e 2 (Use your calculator.)

6. Use the ratio test to show ∑∞

n=1

n!

nn is convergent. Hint: You may need the limit definition of e: e := lim

n→∞ (1 + 1/n)

n .

7. If we know ∑∞

n=0 an, with an > 0, converges what can we say about

∑∞ n=0

1/an? Justify your conclusion.

8. Come up with a counter example to each statement. That is: each statement is false, provide a specific example of it being false.

(a) ∑

(an − bn) = ∑ an −

∑ bn

(b) ∑ anbn =

∑ an ∑ bn

(c) ∑√

an = √∑

an

9. Determine whether the series is absolutely convergent, conditionally convergent or divergent using an appropriate test:

(a) ∑∞

n=1

10n

n!

(b) ∑∞

n=1

(−1)n

n4

(c) ∑∞

n=3

(−1)n

ln √ n

(d) ∑∞

n=1

(−1)n2n

n6

(e) ∑∞

n=1

cos(πn)

n

(f) 1 + 1

2 √

2 +

1

3 √

3 +

1

4 √

4 + . . .

(g) 1

(2 ln 2)2 +

1

(3 ln 3)2 +

1

(4 ln 4)2 +. . .

(h)

∞∑ k=1

√ 2k − 1

4k2 + 1