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# 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 the sequence In converges, 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.

The convergence of the sequence implies the convergence of the integral 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.

# Use integration by parts to prove the functional equation of the Gamma function

Recall the definition of the Gamma function: Using integration by parts, prove the functional equation: Then use mathematical induction to prove that for positive integers we have Incomplete.

# Prove that the improper integral ∫ f(x) dx and ∑ f(n) both converge or both diverge

1. Assume that is a monotonically decreasing function for all and that Prove that the improper integral and the series both converge or both diverge.

2. Give a counterexample to the theorem in part (a) in the case that is not monotonic, i.e., find a non-monotonic function such that converges but diverges.

Incomplete.

# Prove some properties of the improper integrals ∫ (sin x) / x and ∫ (cos t) / t2

1. Prove that the following improper integral converges: 2. Prove that 3. Determine the convergence or divergence of the improper integral 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.