Series

Arithmetic Series

a, a + d, a + 2d, a + 3d, . . . . . . a + (n-1)d

Initial term = a

Difference between successive terms = d

nth Term, $a_n = a + (n-1)d$ 

\[a_i = \frac{a_{i-1} + a_{i+1}}{2}\]

Sum of the first n terms:

\[ S_n = \frac{n(a+a_n)}{2} = \frac{n}{2}[2a+(n-1)d]\]

Geometric Series

$a, \; ar, \; ar^2, \;  . . . . \; ar^{n-1}$

nth Term, $a_n = a r^{n-1}$ 

Sum of the first n terms:

\[ S_n = \frac{a(r^n-1)}{(r-1)} \]

Sum of the infinity terms:

\[ S = \frac{a}{(1-r)}, \;\; |r| < 1 \]

Some Series

\[ 1 + 2 + 3 +4 + . . . . + n = \frac{n(n+1)}{2}\]

\[2 + 4 + 6 + . . .. +2n = n(n+1)\]

\[1 + 3 + 5 + . . . . .+(2n-1) = n^2\]

\[ 1^2 + 2^2 + 3^2  + . . . . + n^2 = \frac{n(n+1)(2n+1)}{6}\]

\[ 1^3 + 2^3 + 3^3  + . . . . + n^3= \left [ \frac{n(n+1)}{2} \right ]^2 \]

\[ 1 + \frac{1}{2} ++ \frac{1}{4} + \frac{1}{8}+ . . . . + + \frac{1}{2^n}+ . . .  = 2\]

Convergence Tests

The Comparison Tests:

Let $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} b_n$ be series such that $0 < a_n < b_n$  for all n, then

• If $\sum_{n=0}^{\infty} b_n$ is convergent then $\sum_{n=0}^{\infty} a_n$ is also convergent.

• If $\sum_{n=0}^{\infty} a_n$ is divegent then $\sum_{n=0}^{\infty} b_n$ is also divergent.

The Ratio Tests:

• If $\lim_{n\rightarrow \infty} \left | \frac{a_{n+1}}{a_n} \right | < 1$ then $\sum_{n=0}^{\infty} a_n$ is convergent.

• If $\lim_{n\rightarrow \infty} \left | \frac{a_{n+1}}{a_n} \right | > 1$ then $\sum_{n=0}^{\infty} a_n$ is divergent.

• If $\lim_{n\rightarrow \infty} \left | \frac{a_{n+1}}{a_n} \right | = 1$ then $\sum_{n=0}^{\infty} a_n$  may converge or diverge and the ratio test is inconclusive; some other test must be used. 

p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges for p > 1 and diverges for 0< p < 1.

Power Series Expansions for Some Functions 

\[e^x = 1 + x + \frac{x^2}{2!}+ \frac{x^3}{3!}+ . . . + \frac{x^n}{n!}+ . . . . \]

\[a^x = 1 + x \ln a + \frac{(x \ln a)^2}{2!}+ \frac{(x \ln a)^3}{3!}+ . . . + \frac{(x \ln a)^n}{n!}+ . . . . \]

\[\ln(1+x) = x - \frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+......+\frac{(-1)^n x^{n+1}}{(n+1)}+ . . ., \;\;-1 <x <1\]

\[\cos(x) = 1 - \frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+......+\frac{(-1)^n x^{2n}}{(2n)!}+ . . .\]

\[\sin(x) = x - \frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+......+\frac{(-1)^n x^{2n+1}}{(2n+1)!}+ . . .\]

\[\tan(x) = x - \frac{x^3}{3}+\frac{2x^5}{15}-\frac{17 x^7}{315}+...... . . ., \;\; |x| < \frac{ \pi}{2}\]

\[\cosh(x) = 1 +\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+......+\frac{ x^{2n}}{(2n)!}+ . . .\]

\[\sinh(x) = x + \frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+......+\frac{ x^{2n+1}}{(2n+1)!}+ . . .\]

\[ \frac{1}{1-x} = 1 +x +x^2+x^3+x^4+...., \;\; |x| <1\]

\[ \frac{1}{1+x} = 1 -x +x^2-x^3+x^4-x^5+...., \;\; |x| <1\]