# 42 Lecture

## MTH101

### Midterm & Final Term Short Notes

## Infinite Series

Infinite series play a crucial role in Calculus and analytical geometry, as they allow us to express functions as infinite sums of simpler functions.

**Important Mcq's**

Midterm & Finalterm Prepration

Past papers included

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**Which of the following tests can be used to determine if an infinite series converges or diverges?**

a) Limit comparison test

b) Ratio test

c) Integral test

d) All of the above

Answer: d) All of the above

**Which of the following series is divergent?**

a) 1 + 1/2 + 1/4 + 1/8 + ...

b) 1 + 1/3 + 1/5 + 1/7 + ...

c) 1/2 + 1/4 + 1/6 + 1/8 + ...

d) 1 - 1/2 + 1/3 - 1/4 + ...

Answer: a) 1 + 1/2 + 1/4 + 1/8 + ...

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

a) Integral test

b) Ratio test

c) Alternating series test

d) Divergence test

Answer: b) Ratio test

**Which of the following series is convergent?**

a) 1 - 1/2 + 1/4 - 1/8 + ...

b) 1 + 1/2 + 1/3 + 1/4 + ...

c) 1 + 1/4 + 1/16 + 1/64 + ...

d) 1/2 + 1/3 + 1/4 + 1/5 + ...

Answer: a) 1 - 1/2 + 1/4 - 1/8 + ...

**What is the nth-term test for divergence?**

a) The series diverges if the limit of the nth term as n approaches infinity is zero.

b) The series converges if the limit of the nth term as n approaches infinity is zero.

c) The test can only be used for series with alternating terms.

d) The test can only be used for series with positive terms.

Answer: a) The series diverges if the limit of the nth term as n approaches infinity is zero.

**Which of the following tests can be used to determine the convergence of an alternating series?**

a) Divergence test

b) Ratio test

c) Integral test

d) Alternating series test

Answer: d) Alternating series test

**Which of the following series is divergent?**

a) 1 - 1/3 + 1/5 - 1/7 + ...

b) 1 + 2 + 3 + 4 + ...

c) 1/2 + 1/3 + 1/5 + 1/7 + ...

d) 1/2 + 1/4 + 1/8 + 1/16 + ...

Answer: b) 1 + 2 + 3 + 4 + ...

**Which of the following tests should be used to determine the convergence of a series with alternating signs and decreasing absolute values?**

a) Divergence test

b) Ratio test

c) Integral test

d) Alternating series test

Answer: d) Alternating series test

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

a) Ratio test

b) Alternating series test

c) Integral test

d) Divergence test

Answer: c) Integral test

**Which of the following series is divergent?**

a) 1/ln(n)

b) 1/n^2

c) 1/n!

d) 1/2^n

Answer: d) 1/

**Subjective Short Notes**

Midterm & Finalterm Prepration

Past papers included

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**What is an infinite series?**

An infinite series is a sum of an infinite number of terms. It is represented as a1 + a2 + a3 +… where a1, a2, a3, … are the terms of the series.

**What is the difference between a sequence and a series?**

A sequence is a list of numbers in a specific order, while a series is the sum of these numbers.

**What is a convergent series?**

A convergent series is a series whose sum approaches a finite value as the number of terms increases.

**What is a divergent series?**

A divergent series is a series whose sum approaches infinity or negative infinity as the number of terms increases.

**What is the nth term test for divergence?**

The nth term test for divergence is a test used to determine if a series converges or diverges by checking if the limit of the nth term as n approaches infinity is zero or not.

**What is the comparison test for convergence?**

The comparison test for convergence is a test used to determine if a series converges or diverges by comparing it to a series that is known to converge or diverge.

**What is the ratio test for convergence?**

The ratio test for convergence is a test used to determine if a series converges or diverges by checking the limit of the ratio of successive terms as n approaches infinity.

**What is the integral test for convergence?**

The integral test for convergence is a test used to determine if a series converges or diverges by comparing it to the integral of a related function.

**What is the alternating series test for convergence?**

The alternating series test for convergence is a test used to determine if an alternating series converges or diverges by checking if the absolute value of the terms decreases and approaches zero.

**What is the limit comparison test for convergence?**

The limit comparison test for convergence is a test used to determine if a series converges or diverges by comparing it to a series whose limit as n approaches infinity is known.

### Infinite series

In Calculus, an infinite series is a sum of an infinite number of terms. These series can either converge, meaning they approach a finite sum, or diverge, meaning they do not approach a finite sum. Infinite series play a crucial role in Calculus and analytical geometry, as they allow us to express functions as infinite sums of simpler functions. In this article, we will discuss the concept of infinite series, their convergence and divergence, and some common types of infinite series. Let us start by defining an infinite series. Suppose we have a sequence of numbers {an}. The sum of this sequence up to the nth term can be denoted by Sn, where Sn = a1 + a2 + ... + an. An infinite series is the sum of an infinite number of terms in a sequence and can be written as S = a1 + a2 + a3 + .... The infinite series S can either converge or diverge. A series converges if its partial sums, that is, the sums of the first n terms of the series, approach a finite limit as n approaches infinity. In other words, if Sn approaches a finite limit as n approaches infinity, then the infinite series S converges, and its sum is the limit of Sn as n approaches infinity. Mathematically, we can write it as S = lim n?? Sn. On the other hand, a series diverges if its partial sums do not approach a finite limit as n approaches infinity. In other words, if Sn does not approach a finite limit as n approaches infinity, then the infinite series S diverges, and it does not have a finite sum. Let us now look at some common types of infinite series. The first type is a geometric series, which is a series of the form a + ar + ar^2 + ... + ar^n-1 + ... , where a and r are constants. The geometric series converges if the absolute value of r is less than 1 and diverges otherwise. In particular, if |r| < 1, then the sum of the geometric series is given by S = a/(1-r). Another type of infinite series is a telescoping series, which is a series in which most of the terms cancel out, leaving only a few terms at the beginning and end of the series. For example, consider the series 1/2 + 1/3 + 1/4 + ... + 1/n, where n is a positive integer. We can write this series as (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n-1 - 1/n) + 1/n, which simplifies to 1/2 - 1/n, which approaches 1/2 as n approaches infinity. Therefore, the series converges to 1/2. The last type of infinite series we will discuss is a power series, which is an infinite series of the form ?an(x-c)^n, where an are constants, x is a variable, and c is a constant. Power series can be used to represent functions as infinite sums of simpler functions. For example, the power series ?(1/n)x^n represents the natural logarithm function ln(1+x) for |x| < 1.**In conclusion,**infinite series are an essential concept in Calculus and analytical geometry. They allow us to express functions as infinite sums of simpler functions and play a crucial role in mathematical analysis. Understanding the convergence and divergence of infinite series is essential in many fields, such as physics, engineering, and economics. The types of infinite series discussed in this article are just a few examples of the many types of infinite series that exist.