Sequence Series
Arithmetic Progression (AP)
General term: $$a_n=a_1+(n1)d$$
Sum of n terms: $$S_n=\frac{n}{2}(a_1+a_n)$$
Properties:
 The difference between two consecutive terms of an AP is constant.
 The sum of the first n odd natural numbers is a perfect square.
 The sum of the first n even natural numbers is n times the (n+1)th odd natural number.
Geometric Progression (GP)
General Term:$$T_n=ar^{n1}$$ Sum of n terms:$$S_n=\frac{a(r^n1)}{r1}, where r\neq1$$
Properties:
 The ratio between two consecutive terms of a GP is constant.
 The sum of the first n terms of a GP with common ratio r and first term a is ( \frac{a(1r^n)}{1r} ), where (r \ne 1).
 If (0 < r < 1), then the sum of the infinite geometric series (a+ar+ar^2+\dotsb) is ( \frac{a}{1r},).
Harmonic Progression (HP)
General term: $$h_n=\frac{1}{a+nd}$$
Sum of n terms: $$S_n= \frac{n}{2a+(n1)d}$$
Series
Convergent series: A series is said to be convergent if it has a finite number of the sums.
Divergent series: A series that doesn’t converge is called a divergent series.
Alternating series: An alternating series is a series whose terms alternate in sign.
Mathematical induction: Mathematical induction is a method of proving that a statement holds for all natural numbers.
Limits of sequences
Definition of limit: The limit of a sequence ((a_n)) is the number L if, for any (\varepsilon>0), there exists an integer (N) such that (a_n  L< \varepsilon) whenever (n>N).
Properties of limits:

If (\lim\limits_{n\to \infty}a_n = L) and (\lim\limits_{n\to\infty}b_n = M,) then a) (\lim\limits_{n\to\infty} (a_n +b_n) = L + M). b) (\lim\limits_{n\to\infty} (a_n  b_n) = L  M). c) (\lim\limits_{n\to\infty} (ca_n) = cL, where c is a constant). d) If (c\neq0), (\lim\limits_{n\to\infty} \frac{a_n}{b_n}=\frac{L}{M}).

If (\lim\limits_{n\to \infty}a_n = L), then for every (\varepsilon > 0), there exists an integer (N) such that (a_nL\leq\varepsilon) for all (n>N).
Limit theorems:

Squeeze theorem: If (a_n \le b_n \le c_n) for all (n) and if (\lim\limits_{n\to \infty} a_n = \lim\limits_{n\to \infty} c_n = L,) then (\lim\limits_{n\to \infty} b_n = L.)

L’Hopital’s rule: If (\lim\limits_{n\to a}f(x) = \lim\limits_{n\to a} g(x) = 0) or (\pm \infty), and if (\lim\limits_{n\to a} \frac{f’(x)}{g’(x)}) exists, then (\lim\limits_{n\to a} \frac{f(x)}{g(x)} = \lim\limits_{n\to a} \frac{f’(x)}{g’(x)}).
Applications of sequences and series

Approximating functions: Sequences and series can be used to approximate functions. For example, the Maclaurin series for (e^x) is $$e^x = \sum\limits_{n=0}^{\infty}\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+ \dots$$ This series can be used to approximate the value of (e^x) for any real number (x).

Calculating sums: Sequences and series can be used to calculate the sums of infinite series. For example, the sum of the series $$\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1}+\frac{1}{2^2}+\frac{1}{3^2}+\dots$$ can be calculated using the formula $$\sum\limits_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$

Solving equations: Sequences and series can be used to solve equations. For example, the equation (x = e^x) can be solved using the iterative method: $$x_{n+1} = e^{x_n}$$ Starting with an initial guess (x_0), we can use this formula to generate a sequence of approximations that converge to the solution of the equation.