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Saturday, 4 July 2020

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


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