Read Series

Sequence is nothing but a specific order (succession of numbers or things or objects) which the terms present in the list follow one after other, for example sequence of even numbers. A sequence might be finite or infinite depenting upon the number of terms present in it. 

It may be finite if the number of terms present is finite, else, infinite. On adding the sum of the sequence, we can obtain the series. For example 1,3,5,7 .. is a sequence of odd numbers., Where as series is represented as 1+3+5+5+7+..... Learning about series is explained with examples in the following sections. The following content helps a lot to learn and read about series

      

Having problem with Types of Series keep reading my upcoming posts, i will try to help you.

Impotant terms in learning series:

As said earlier, series is the sum of the terms in the sequence.The general representation for sum of a series is given by the expression,  
                    
sum_(k=1)^n` a^k

Some important formulas have been derived to make learning series much easier. These formulas are the most frequently used kind of series,

1. 1+ 2 +3 + 4 + 5 +....+ n = `1/2` (n(n+1))

2. 1²+ 2² +3² + 4² + 5² +....+ n² =`1/6` n(n+1)(2n+1)

3. 1³+ 2³ +3³ + 4³ + 5³ +....+ n³ = [`1/2` (n(n+1)]²

4. 1³+3³ + 5³ +....+ (2n-1)³ = n² (2n² -1)

Example problems to read series.:

Example 1 to read series:
Find the sum of n terms of the series whose nth term is n(n+3)
Solution:

Sn = `sum_(k=1)^n` a_k

      = `sum_(k=1)^1` (k² +3k) = ^{`sum_(k=1)^1`} k²  + `sum_(k=1)^1` 3k

     = `1/6` n(n+1)(2n+1) +3.`1/2` (n(n+1)

     = `1/6` [ n(n+1)(2n+1) +9n(n+1)]

     = `1/6` [ n(n+1)(2n+1+9)]

     = `1/6` n(n+1).2(n+5)

     = `1/3` n(n+1)(n+5)

Hence the required sum is = `1/3` n(n+1)(n+5)

I am planning to write more post on Alternate Interior Angle Definition and samacheer kalvi syllabus. Keep checking my blog.

Example 2 to read series:
Find the sum of the first ten numbers form 1 to 10,

Solution:

The formula for finding the sum of the first ten numbers in the sequence 1, 2, 3,.......10.

1+ 2 +3 + 4 + 5 +....+ n = `1/2` (n(n+1))

 = `1/2` (10(10+1))

 = `1/2` (10(11))

 = `1/2` (110)

 = 55