Show that ∞ X n=1 1 n2 is convergent using other method

1. Midterm 1:Sample Exam (1) It is easy to say that

∞

X

n=1

1

n² is convergent by p-test because p = 2 > 1. Show that

∞

X

n=1

1

n² is convergent using other method.

(2) Convergence, divergence? Explain.

(a)

∞

X

n=1

(−1)ⁿ⁻¹ n^1/4 (b)

∞

X

n=1

n²+ n n⁵+ 3n + 2 (c)

∞

X

n=1

 2n + 3 4n − 1

n

.

(d)

∞

X

n=1

n!n!

(2n)!.

(3) Absolutely convergent? Conditionally convergent? Divergent?

∞

X

n=1

(−1)ⁿ⁻¹(√ⁿ n − 1).

(4) Compute lim

x→4

4 − x 5 −√

x²+ 9 (5) Compute the limit

(a) lim

x→−2

px²+ 3x + 6, (b) lim

x→2

√6 − x − 2

√3 − x − 1. (c) lim

x→0

tan 2x sin 5x. (6) Suppose x_n= cosx

2cos x

2²· · · cos x

2ⁿ. Here x is a real number. Find lim

n→∞x_n. (7) Bonus....

1