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Line 1: − In mathematics a '''series''' is the sum of a [[sequence]] of numbers. This article is intended to give the reader some understanding on the summation of finite series and why some infinite series converge whilst others diverge.+ − − ==Summation notation== − If ever value in a sequence is denoted <math>a_{n}</math>, then the sum of ''N'' of these are denoted, − − <math>a_{1}+a_{2}+a_{3}+\dots+a_{N}=\sum^{N}_{n=1}a_{n}</math> − − This is usually pronounced "the sum of 1 to N of a n" (note that this is still an ambiguous statement which is why mathematics is more rigorous written down). − − If we are concerned with particular number we can move the starting index. For example, − − <math>\sum_{n=500}^{1000}n</math> − − is the sum of the integers from 500 to 1000. − − ==Finite series== − ''For more detail see [[Finite series]].'' − − The sum of finite series is in general relatively straight forward to calculate it consist of merely summing together all the numbers in a sequences. However this can be time consuming and in pre-computer mathematics many shortcuts were found. − − Gauss once famously was given the assignment by his teacher to add the number between 1 and 100. The rest of the students were working hard adding carefully, but after less than a minute he correctly wrote 5050. This story whilst is allegorical leads to a key insight in finite sums, − − :<math>\sum^{N}_{n=1}n=\frac{N(N+1)}{2}</math> − − ==Infinite series== − ''For more detail see [[Infinite series]].'' − − The idea that an infinite number of additions lead to a finite number was initial seen as problematic or even impossible, (see [[Zeno's paradox]]). However it is straightforward to see why this could happen, − − :Let <math>S=\sum^{\infty}_{n=1}\frac{1}{2^{n}}</math> − − :<math>S=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots</math> − − :<math>2S=1+\frac{1}{2}+\frac{1}{4}+\dots</math> − − :<math>2S=1+S</math> − − :<math>S=1</math> − − This one of an important class of infinites series called the geometric series. − − ===Geometric series=== − − <math>\sum^{\infty}_{n=1}r^{n}=\frac{1}{1-r},\qquad \mathrm{for}\ |r|<1.</math> − − Where <math>|r|</math> denotes the [[absolute value]] of ''r''. − − ===Convergence=== − ''For more detail see [[Convergence]]'' − − However all infinite series don't converge, a famous example is, − − :<math>\sum^{\infty}_{n}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\dots</math> − − :<math>\geq 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{4}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\dots</math> − − :<math>=1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\dots</math> − − Adding together an infinite string of halves will not give you a finite number so this sequence is said to '''diverge'''. − − A series will only converge if <math>\lim_{n\rightarrow\infty}a_{n}=0</math>. However there are two other conditions for real series, − − ====Power Series convergence==== − − A power series is of the form, − − :<math>\sum^{\infty}_{n=1}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-a)^{n}</math>, − − This will converge if <math>\lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1</math>. − − ====Alternating Series convergence==== − − An alternating series is of the form, − − :<math>\sum^{\infty}_{n=1}(-1)^{n}a_{n}</math> − − This will converge if <math>0\leq a_{n+1}<a_{n}</math>. − − ==Usage== − *[[Integral]] − *[[Riemann Zeta function]] − *[[Taylor series]] − *[[Fourier series]] − − [[Category:mathematics]]
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