### Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

# Arithmetico-geometric series

Suppose is an A.P and is in G.P, then is known as an Arithmetico-geometric sequence.

Accordingly is called an Arithmetic-geometric series.

An Arithmetic-geometric sequence is of the form,

[where and are respectively common difference and common ratio corresponding to the arithmetic series and the geometric series ].

Partial Sum

Sum of terms of an Arithmetic-geometric series is,

or

For < i.e < < , then and .

Therefore, the sum of an infinite number of terms of the sequence is,

For which the series is convergent.

Some Different Kinds of Series -:

(i)  Binomial Series -:

We know from binomial theorem for positive integral index , that,

.

The above expression holds for any value of .

Binomial Theorem for a Real Index

where , provided < .

This is known as the Binomial Series and it has sum for < .

If , the binomial series becomes the finite series and has the sum without any restriction on .

Series Expansion of a Function -:

If a series has a finite sum we say that the series is convergent and write

If is the sum of a series, the series is said to be an expansion of .

As is an expansion of

Application of Binomial Series

(i)  ,for <

(ii) , for <

(iii) , for <

(iv) , for   <

(v) , for

(vi) , for

(vii) , for

(viii) , for

The Exponential Series -:

We know,

If is any real number, by the binomial theorem for real index we get, for .

Since , taking the limit as , we get , for

This series is called Exponential Series and also called the expansion of .

Let and

i.e.

(Putting in place of )

i.e.

This series is also called the Exponential series and called the expansion of .

The Logarithmic Series -:

Let so that ,then

Restricting so that < , then

From above equations, we have < .

This series is called the Logarithmic series and also called the expansion of .

<

< .

Above two expressions are the expansion of and .

Some Finite Series -: (For terms)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii.

(viii)

Some Infinite Series -:

(i)

(ii)

(iii)

Convergence of Infinite Series -:

, if

Properties of Convergent Series -:

Convergent Series -:

(i)

(ii)

Convergence Tests -:

Let and be series such that and are positive for all ,i.e. < .

(i) If is convergent, then is also convergent.

(ii) If is divergent, then is also divergent.

The Limit Comparison Test -:

Let and be series such that  and are positive for all .

(i) If   < then and are either both convergent or both divergent.

(ii) If , then is convergent implies that is also convergent.

(iii) If , then is divergent implies that is also divergent.

Absolute Convergent -:

A series is absolutely convergent if the series is convergent.

If the series series is absolutely convergent then it is convergent.

Conditional Convergent -:

A series is conditionally convergent if the series is convergent but it is not absolutely convergent.

The Ratio Test -:

Let be a series with positive terms.

(i) If < , then is convergent.

(ii) If , then is divergent.

(iii) If , then may converge or diverge and the ratio test is inconclusive.

The Root Test -:

Let be a series with positive terms.

(i) If , then is convergent.

(ii) If , then is divergent.

(iii) If , then may converge or diverge, but no conclusion can be drawn from this test.

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