**Introduction
to Binomial, Exponential and Logarithmic series**

** **

**Binomial
series**

A
binomial is an algebraic expression of two terms. Now let us see the following
binomial expansion and the number pattern we get adjacent to it.

From
the above, we observe that the binomial coefficients form a number pattern
which is in a triangular form. This pattern is known as Pascal’s triangle. [ In
pascal’s triangle, the binomial coefficients appear as each entry is the sum of
the two above it.]

** **

**Binomial theorem for a positive
integral index:**

For
any natural number n

In
the above binomial expansion, we observe,

(i) The (*r*+1^{)th} term is denoted by *T *_{r}_{+1}* *=* nc*_{r} x^{n}^{–}^{r}*a*^{r}

(ii)
The degree of ‘*x*’ in each term decreases while
that of ‘*a*’ increases such that the sum of the power in each term
equal to *n*

(iii)* nc*_{0},* nc*_{1},* nc*_{2}, …* nc*_{r}, …* nc*_{n} are binomial coefficients they are
also written as* c*_{0},* c*_{1},* c*_{2}* *… ,* c*_{n}

(iv)
From the relation *nc*_{r} = *nc*_{n}_{–}_{r} we see that the coefficients of
term equidistant from the beginning and the end are equal.

** **

**Example 7.15**

Expand
(2*x*+*y*)^{5} using binomial theorem.

*Solution:*

** **

**Example 7.16**

Find
the middle terms of expansion (3*x*+*y*)5

*Solution:*

In
the expansion of (3*x*+*y*)5 we have totally 6 terms. From
this the middle terms are T_{3} and T_{4}

Similarly
we can find by putting *r* = 3 in *T*_{r}_{+1} to get *T*_{4} , then
*T*_{4} =90*x*^{2}*y*^{3}

**Binomial theorem for a rational
index:**

For
any rational number other than positive integer

** **

**Some important expansions:**

** **

**Special cases of infinite series:**

** **

**Example 7.17**

Find the approximate value of ^{3}√1002 (correct to 3 decimal places) using Binomial
series.

*Solution:*

** **

**Exponential
series**

** **

**Example 7.18**

*Solution:*

** **

**Logarithmic
series:**

** **

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

11th Statistics : Chapter 7 : Mathematical Methods : Introduction to Binomial, Exponential and Logarithmic series |