Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Series and types of Series

Series -:

 \Sigma t_n = t_1+t_2+t_3...+ t_n+...\  , where  t_n is the  n^{th} term of a sequence, is called a Series.

 \to If the number of terms is finite, then it is called Finite Series and if the number of terms is infinite, it is called Infinite Series.

Finite series -:

For a given  n\in N,\ \sum_{k=1}^{n} t_k=t_1+t_2+...+t_n , is a Finite Series.

Infinite Series -:

An expression the  \sum_{n=1}^{\infty }t_n=t_1+t_2+t_3+... which involves addition of infinitely many real numbers, which we clarify to find the the sum of infinitely many real numbers.

Partial Sums of an Infinite Series -:

For an infinite series  \sum_{n=1}^{\infty }t_n , a sum  S_n = \sum_{k=1}^{n} t_k is called the  n^{th} partial sum of the series;

for  n= 1,2,3,...

Thus  S_1=t_1,\ S_2= t_1+t_2,...,

 S_n=t_1+t_2+...+t_n , and so on.

Consequently, given a sequence  (t_n)_{n=1}^\infty , we obtain another related sequence   (S_n)_{n=1}^\infty  , which is the sequence of  n^{th} partial sums of the infinite series  \sum_{n=1}^{\infty }t_n .

Sum of an Infinite Series -:

Let  (S_n) be the sequence of partial sums of an infinite series  \Sigma t_n .

 (S_n) is convergent, if there exists  s\in R , such that  \lim_{n \to \infty }  S_n=s .

 \to If  (S_n) is convergent, then  \Sigma t_n is said to be convergent and  s is called its sum.

 \to  \Sigma t_n is convergent, if the sequence  (S_n) of partial sums is convergent.

\to If  (S_n) does not converge, i.e.  \lim_{n \to \infty } S_n does not exist, then the series  \Sigma t_n is called a Divergent Series, and we say  \Sigma t_n diverges.

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