Finite series

A finite series is a summation of a finite number of terms.

Sum of first n natural numbers:

$$ 1+ 2+ 3+\cdots = \frac {n(n+1)}{2} $$

Example:

Find the sum of first 20 terms of an A.P.

Solution:

S_n = n(n+1) / 2

S_n = 20(20+1)/2

S_n = 10x21

S_n = 210

Sum of Even Numbers:

$$ 2+4+6+\cdots = n(n+1) $$

Example:

What is the sum of even numbers from 1 to 60?

Solution:

S_n = n(n+1)

S_n = 60(60+1)

S_n = 60 x 61

S_n = 3660

Sum of Odd Numbers:

$$ 1+3+5+\cdots = n^2 $$

Example:

What is the sum of odd numbers from 1 to 50?

Solution:

We know that, from 1 to 50, there are 25 odd numbers.

Thus, n = 25

S_n = n²

S_n = 25² = 625

S_n = 625

Sum of the first n squares:

$$ \sum n^2 = \frac {n(n + 1)(2n + 1)}{6} $$

Example:

Find the addition of squares of the first 30 natural numbers.

Solution:

Here, n = 30

Σ30² = (30/6) (30 + 1)(2 x 30 + 1)

Σ30² = (5)(31)(61)

Σ30² = 9455

Sum of Squares of First n Odd Numbers:

$$ 1^2 + 3^2 + 5^2 + \cdots = \frac {n(4n^2-1)}{3} $$

Sum of the first n cubes:

$$ 1^3 + 2^3 + 3^3 + \cdots = \frac {n^2(n + 1)^2}{4} $$

Sum of cube of first n odd natural numbers:

$$ 1^3 + 3^3 + 5^3 + \cdots = n^2(2n^2 - 1) $$

Example:

Find the sum of cube of first 4 odd numbers

Solution:

Sum of cube of first 4 odd numbers = 1³ + 3³ + 5³ + 7³

put n = 4

= n²(2n² - 1)

= 4²(2x(4)² - 1)

= 16(32-1)

= 496

Other Series:

$$ 1+\frac 12+\frac 14+\cdots =2 $$

$$ 1+\frac {1}{1!}+\frac {1}{2!}+ \cdots =e $$