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

Sequence

A real sequence is a function whose domain is the set of positive integers and range is a subset of the set of real numbers, i.e.,  f: N \longrightarrow R

Example:

Let  f(n)=\dfrac{1}{n}, \forall n \in N , then  f is a sequence in  R .

Limit of Sequence:

Definition:

We say that a sequence of real numbers  (a_n)^\infty_{n-1} converges to a real number  l   if for every  \in > 0 there is an   N_0 such that  |a_n-l| < \in, \forall n > N_0 i.e., a sequence  (a_n)^{\infty}_{n-1} converges at all it has to converge to a unique real number  l which we call its limit, we express this by

 \lim_{n \to \infty}  a_n=l

where the symbol  '\longrightarrow ' stands for approaches or tends to.

Properties:

If  \lim_{n \to \infty}  a_n=l_1 and  \lim_{n \to \infty} b_n=l_2  then,

  •  \lim_{n \to \infty} (a_n+b_n)=l_1+l_2=\lim_{n \to \infty} a_n+\lim_{n \to \infty} b_n
  •  \lim_{n \to \infty}(a_n-b_n) =l_1-l_2=\lim_{n \to \infty} a_n-\lim_{n \to \infty} b_n
  •  \lim_{n \to \infty} (a_n \cdot b_n)=l_1\cdot l_2=\lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n
  •  \lim_{n \to \infty} \left( \dfrac{a_n}{b_n} \right)=\left( \dfrac{l_1}{l_2} \right)=\dfrac{\lim_{n \to \infty}a_n}{\lim_{n \to \infty}b_n }

Where  \lim_{n \to \infty} b_n \neq 0

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