# Exponential and Logarithmic Series

#### Have you ever noticed the use of the word exponential in our daily life?

- It is used to
**express something which****is rising or increasing at a steady and a usual rapid rate**or**something which increases quickly by large amounts**.

*Examples***:**

(i) Prices have increased exponentially i.e. at an exponential rate.

(ii) The budget is increasing at an exponential rate.

(iii) We all want to progress exponentially and achieve success in our lives.

- Something is said to increase or decrease exponentially if its rate of change must be expressed using exponents.
- A graph of such a rate would appear not as a straight line, but as a curve that continually becomes steeper or shallower.
- In Math, it refers to a Mathematical (function, curve, series, or equation) containing, or involving one or more numbers or quantities raised to an exponent.

So, here we try to identify what does **Exponential series** mean in mathematics.

__Sequence__

A sequence is a list or arrangement of numbers in a definite order written according to some rule or pattern.

__Series__

If the *n* terms of the sequence are *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4}, ……., *a _{n}* then the expression

*a*

_{1}+

*a*

_{2}+

*a*

_{3}+

*a*

_{4}+ ……. +

*a*is called the series associated with the given sequence.

_{n}

__Exponential Series__* *

**1. A series of the form**

is called the **Exponential series, x ∈**

**Here**

*R*.*e*is called the

^{x}**exponential function.**

**Replace x by –x **in the above series given by (1), we obtain

= 1 + ∞ =

**2. =** 1 – ∞ =

**On adding (1) and (2) we have**

** + = **2

**3. ** = 1 + ∞

**Subtracting (2) from (1),**

= 2

**4. ** *= x + ∞ = *

__Particular cases:__

**Put x = 1 in (1) and (2), **

**we have**

**5. ** ∞ =

Here *e* is an irrational number such that 2 < *e* < 3.

The value of *e* = 2.7182818284

**6. ∞ = **

**Put x = 1 in (3) and (4) we have**

**7. ∞ = **

**8. ∞ = **

__Logarithmic Series__

**The logarithmic series**is defined for |*x*| < 1 and is given by

*(1+x) = x – ∞*

**Replace ***x* by –*x* in the above series, we obtain

*(1 – x) = – x – ∞ or*

–*(1-x) = x + ∞*

**On adding (1) and (2)** we have

*(1+x) + (1-x) = (1-) = -2 , |x|< 1*

**Subtracting (2) from (1), **we have

*(1+x)-**(1-x) =*= 2

**Put x = 1 in (1), we have**

- 2=1 – ∞ =∞

**Note:** log* _{e}*(1 +

*x*) is defined for

*x*= 1.

__Examples__

Now let’s consider some examples on exponential and logarithmic series.

** Example 1:** Find the value of 1+ ∞ .

Since, ** = **1 + ∞.

Hence, put *x* = 5 in above expression, we have

= 1 + ∞

** Example 2:** Find the value of 1 – ∞

Since, = 1 – ∞

Hence, put *x* = 2.5 in above expression, we have

= = 1 – ∞

** Example 3:** Find the value of ∞ .

∞ = ∞

=

=+=*e *

** Example 4:** Find the value of (0.5) – +–+……….∞

Since, *(1+x) = x – ∞*

Hence, put *x *= 0.5 we have

*(1+0.5) = (0.5) – ∞*

log = log = (0.5) – *∞*

** Example 5:** Find the value of .∞.

Since, *(1-x) = -x – ∞*

–*(1-x) = x + ∞*

Hence, put *x= * we have

#### Check Point

- Find the value of ∞
- Find the value of ∞
- Find the value of ∞
- Find the value of ∞
- Find the value of ∞

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