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# Find the angle between the vectors (2,4,6, …, 2n) and (1,3,5,…,2n-1) as n goes to infinity

Consider the vectors and in . Let denote the angle between and . Find the value of as .

If is the angle between and then we have

Computing each of these pieces we have

Therefore,

Hence, as which implies as .

# Find the angle between the vectors (1,1, …, 1) and (1,2,…,n) as n goes to infinity

Consider the vectors and in . Let denote the angle between and . Find the value of as .

If is the angle between and then we have

Thus,

Therefore, as .

# Show that the limit and derivative cannot be interchanged when fn(x) = (sin (nx)) / n

For each positive integer and and all real define

Prove that

Proof. First, we have

for all . Hence,

On the other hand,

Hence,

# Show that we cannot interchange a limit and integral for fn(x) = nxe-nx2

For each positive integer define

Prove that the following limit and integral cannot be interchange:

Proof. First, we have

On the other hand,

Hence,

# Prove or disprove: ∫ f(x) converges implies lim f(x) = 0

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

The convergence of the improper integral

implies

Counterexample. The idea of the construction is a function which has rapidly diminishing area, but has a height that is not going to 0. (So, for an idea consider triangles on the real line all with height 1, but for which the base is becoming small rapidly.) To make this concrete, define

for each positive integer . Then for the improper integral we have

which we know converges. On the other hand

for all positive integers . Hence,

(since it does not exist). Hence, the statement is false.

(Note: For more on this see this question on Math.SE.)

# Prove or disprove a statement relating the derivative of a function to an improper integral of the function

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

Assume exists for all and is bounded,

for some constant for all . Then,

Incomplete.

# Prove or disprove: If f is positive and lim In = A, then ∫ f(x) converges to A

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

If is positive and if

then

Incomplete.

# Prove or disprove: If lim f(x) = 0 and lim In = A then ∫ f(x) converges to A

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

Assume

Then

Incomplete.

# Prove or disprove: If f is monotonic decreasing and lim In exists then ∫ f(x) converges

The following function is defined for all , and is a positive integer. Prove or provide a counterexample to the following statement.

If is monotonically decreasing and if

exists, then the improper integral

converges.

Incomplete.

# Prove some properties of improper integrals involving 1/x and sin x

1. Prove the following limit formulas:

2. Determine whether the following improper integrals converge:

Incomplete.