# 43 Lecture

## MTH101

### Midterm & Final Term Short Notes

An alternating series is a series whose terms alternate in sign. The alternating series test states that if an alternating series satisfies the conditions that the absolute value of the terms decreases and approaches zero as the index approaches

## Important Mcq's Midterm & Finalterm Prepration Past papers included

Which of the following tests is used to determine if a series is absolutely convergent?

A) Ratio test

B) Integral test

C) Root test

D) Comparison test

Which of the following series is convergent?

A) ? n=1 to infinity (1/n)

B) ? n=1 to infinity (1/(n^2))

C) ? n=1 to infinity (1/n^3)

D) ? n=1 to infinity (n^2)

Answer: B) ? n=1 to infinity (1/(n^2))

Which of the following convergence tests is based on comparing the given series with a simpler series whose convergence or divergence is known?

A) Root test

B) Ratio test

C) Comparison test

D) Integral test

Which of the following series is divergent?

A) ? n=1 to infinity (1/2^n)

B) ? n=1 to infinity (1/n!)

C) ? n=1 to infinity (n/2^n)

D) ? n=1 to infinity (1/n)

Answer: B) ? n=1 to infinity (1/n!)

Which of the following tests is used to determine if a series is conditionally convergent?

A) Alternating series test

B) Divergence test

C) Integral test

D) Comparison test

Which of the following tests can be used to determine the convergence of a series with positive terms?

A) Divergence test

B) Ratio test

C) Integral test

D) Root test

Which of the following tests is used to determine the convergence of an alternating series?

A) Ratio test

B) Integral test

C) Root test

D) Alternating series test

Which of the following tests can be used to determine the convergence of a series with negative terms?

A) Integral test

B) Comparison test

C) Root test

D) Divergence test

Which of the following series is convergent?

A) ? n=1 to infinity (n/2^n)

B) ? n=1 to infinity (1/n^2 + 2)

C) ? n=1 to infinity (1/ln(n))

D) ? n=1 to infinity (n^(3/2)/(n^2 + 1))

Answer: A) ? n=1 to infinity (n/2^n)

Which of the following convergence tests is used to determine the convergence of a series with non-negative terms, but whose terms do not approach zero?

A) Ratio test

B) Root test

C) Integral test

D) Divergence test

## Subjective Short Notes Midterm & Finalterm Prepration Past papers included

What is the comparison test for convergence of infinite series?

Answer: The comparison test for the convergence of an infinite series is a method of determining whether a given series converges or diverges by comparing it with another series.

What is the ratio test for convergence of infinite series?

Answer: The ratio test is a convergence test for infinite series. It states that if the limit of the ratio of consecutive terms of a series is less than 1, then the series converges absolutely.

What is the root test for convergence of infinite series?

Answer: The root test is a convergence test for infinite series. It states that if the limit of the nth root of the absolute value of the nth term of a series is less than 1, then the series converges absolutely.

What is the integral test for convergence of infinite series?

Answer: The integral test is a convergence test for infinite series. It states that if the integral of the function corresponding to the series converges, then the series converges.

What is the alternating series test for convergence of infinite series?

Answer: The alternating series test is a convergence test for infinite series in which the terms alternate in sign. It states that if the absolute value of the terms decrease monotonically to 0, then the series converges.

What is the alternating series error bound?

Answer: The alternating series error bound is an estimate of the error involved in approximating the sum of an alternating series with a finite number of terms.

What is the Cauchy condensation test for convergence of infinite series?

Answer: The Cauchy condensation test is a convergence test for infinite series. It states that if the terms of a series decrease monotonically to 0, then the series converges if and only if the corresponding series obtained by taking the sum of powers of 2 of the terms converges.

What is the absolute convergence test for infinite series?

Answer: The absolute convergence test is a convergence test for infinite series. It states that if the absolute value of each term of a series converges, then the series converges absolutely.

What is the p-series test for convergence of infinite series?

Answer: The p-series test is a convergence test for infinite series of the form 1/n^p, where p is a positive number. It states that if p > 1, then the series converges; if p <= 1, then the series diverges.

What is the limit comparison test for convergence of infinite series?

Answer: The limit comparison test is a method of determining whether a given series converges or diverges by comparing it with another series. It states that if the limit of the ratio of the terms of the two series is a positive constant, then the two series either both converge or both diverge.

Calculus deals with a lot of infinite series, and while it is relatively easy to determine the convergence or divergence of a series through the use of tests like the comparison test, limit comparison test, and the integral test, sometimes these tests are not enough to determine the convergence or divergence of a series. In such cases, additional convergence tests are needed to make the determination.

### Alternating Series Test:

An alternating series is a series whose terms alternate in sign. The alternating series test states that if an alternating series satisfies the conditions that the absolute value of the terms decreases and approaches zero as the index approaches infinity, then the series converges.

### Ratio Test:

The ratio test is a convergence test that compares the absolute values of consecutive terms in a series. It states that if the limit of the absolute value of the ratio of the (n+1)th term and the nth term is less than 1, then the series converges absolutely. If the limit is greater than 1, then the series diverges, and if the limit is equal to 1, then the test is inconclusive.

### Root Test:

The root test is another convergence test that compares the absolute values of consecutive terms in a series. It states that if the limit of the nth root of the absolute value of the nth term is less than 1, then the series converges absolutely. If the limit is greater than 1, then the series diverges, and if the limit is equal to 1, then the test is inconclusive.

### Comparison Test:

The comparison test is a convergence test that compares the given series with another series whose convergence or divergence is known. If the series being tested is smaller than a convergent series, then it also converges. If it is larger than a divergent series, then it also diverges.

### Limit Comparison Test:

The limit comparison test is a convergence test that is similar to the comparison test but uses limits to compare the given series with another series whose convergence or divergence is known. If the limit of the ratio of the two series is a positive number, then both series converge or both diverge. If the limit is zero or infinity, then the test is inconclusive.

### Integral Test for Convergence:

The integral test for convergence compares a given series with a corresponding improper integral. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

### Absolute Convergence Test:

If a series and its absolute value series both converge, then the series is said to converge absolutely. Absolute convergence implies convergence, but the converse is not always true. Conditional Convergence Test: If a series converges but does not converge absolutely, then it is said to converge conditionally.

### Raabe's Test:

Raabe's test is a convergence test that is used to determine the convergence or divergence of a series whose terms decrease slowly to zero. If the limit of the ratio of the nth and (n+1)th terms approaches 1, then the series may converge or diverge, and the test is inconclusive. If the limit is less than 1, then the series converges, and if the limit is greater than 1, then the series diverges.

### Gauss's Test:

Gauss's test is a convergence test that is used to determine the convergence or divergence of a series whose terms are positive and whose ratio of successive terms approaches 1. If the sum of the first n terms of the series is less than or equal to Kn^2 for some positive constant K and for all n, then the series converges. If the sum of the first n terms of the series is greater than or equal to Kn^2 for