What Does It Mean for a Series to Converge

Mathematical series with a finite sum

In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence ${\displaystyle (a_{0},a_{1},a_{2},\ldots )}$ defines a series S that is denoted

${\displaystyle S=a_{0}+a_{one}+a_{two}+\cdots =\sum _{thousand=0}^{\infty }a_{thousand}.}$

The n th partial sum S _n is the sum of the kickoff due north terms of the sequence; that is,

${\displaystyle S_{n}=\sum _{k=1}^{north}a_{k}.}$

A series is convergent (or converges) if the sequence ${\displaystyle (S_{one},S_{2},S_{three},\dots )}$ of its partial sums tends to a limit; that means that, when adding one ${\displaystyle a_{k}}$ after the other in the order given by the indices, i gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number ${\displaystyle \ell }$ such that for every arbitrarily small positive number ${\displaystyle \varepsilon }$ , in that location is a (sufficiently large) integer ${\displaystyle N}$ such that for all ${\displaystyle north\geq N}$ ,

${\displaystyle \left|S_{n}-\ell \correct|<\varepsilon .}$

If the series is convergent, the (necessarily unique) number ${\displaystyle \ell }$ is called the sum of the serial.

The aforementioned notation

${\displaystyle \sum _{k=1}^{\infty }a_{k}}$

is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of calculation a and b besides as the issue of this addition, which is chosen the sum of a and b.

Any series that is non convergent is said to be divergent or to diverge.

Examples of convergent and divergent series [edit]

Convergence tests [edit]

There are a number of methods of determining whether a serial converges or diverges.

If the blueish series, ${\displaystyle \Sigma b_{n}}$ , tin be proven to converge, so the smaller series, ${\displaystyle \Sigma a_{north}}$ must converge. By contraposition, if the blood-red series ${\displaystyle \Sigma a_{n}}$ is proven to diverge, so ${\displaystyle \Sigma b_{n}}$ must also diverge.

Comparison exam. The terms of the sequence ${\displaystyle \left\{a_{n}\correct\}}$ are compared to those of some other sequence ${\displaystyle \left\{b_{due north}\right\}}$ . If, for all n, ${\displaystyle 0\leq \ a_{n}\leq \ b_{due north}}$ , and ${\textstyle \sum _{n=ane}^{\infty }b_{north}}$ converges, then then does ${\textstyle \sum _{n=one}^{\infty }a_{northward}.}$

Withal, if, for all northward, ${\displaystyle 0\leq \ b_{n}\leq \ a_{north}}$ , and ${\textstyle \sum _{n=ane}^{\infty }b_{due north}}$ diverges, then so does ${\textstyle \sum _{n=ane}^{\infty }a_{northward}.}$

Ratio examination. Presume that for all n, ${\displaystyle a_{north}}$ is not nil. Suppose that there exists ${\displaystyle r}$ such that

${\displaystyle \lim _{n\to \infty }\left|{\frac {a_{n+i}}{a_{n}}}\right|=r.}$

If r < ane, so the series is absolutely convergent. If r > 1, then the serial diverges. If r = i, the ratio test is inconclusive, and the series may converge or diverge.

Root test or northwardthursday root test. Suppose that the terms of the sequence in question are not-negative. Ascertain r every bit follows:

${\displaystyle r=\limsup _{north\to \infty }{\sqrt[{north}]{|a_{n}|}},}$

where "lim sup" denotes the limit superior (perhaps ∞; if the limit exists it is the aforementioned value).

If r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge.

The ratio examination and the root examination are both based on comparison with a geometric series, and as such they work in like situations. In fact, if the ratio examination works (meaning that the limit exists and is not equal to one) then so does the root test; the antipodal, withal, is not true. The root test is therefore more mostly applicable, but equally a practical affair the limit is often difficult to compute for commonly seen types of series.

Integral test. The series tin can be compared to an integral to constitute convergence or departure. Let ${\displaystyle f(due north)=a_{n}}$ exist a positive and monotonically decreasing function. If

${\displaystyle \int _{one}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{ane}^{t}f(x)\,dx<\infty ,}$

then the series converges. But if the integral diverges, and then the series does and then also.

Limit comparison test. If ${\displaystyle \left\{a_{north}\correct\},\left\{b_{n}\right\}>0}$ , and the limit ${\displaystyle \lim _{due north\to \infty }{\frac {a_{due north}}{b_{n}}}}$ exists and is non cipher, and then ${\textstyle \sum _{n=1}^{\infty }a_{n}}$ converges if and only if ${\textstyle \sum _{northward=1}^{\infty }b_{n}}$ converges.

Alternating series test. Also known as the Leibniz benchmark, the alternating series test states that for an alternating series of the grade ${\textstyle \sum _{n=1}^{\infty }a_{n}(-1)^{n}}$ , if ${\displaystyle \left\{a_{due north}\right\}}$ is monotonically decreasing, and has a limit of 0 at infinity, then the series converges.

Cauchy condensation test. If ${\displaystyle \left\{a_{northward}\right\}}$ is a positive monotone decreasing sequence, and then ${\textstyle \sum _{due north=i}^{\infty }a_{north}}$ converges if and only if ${\textstyle \sum _{k=one}^{\infty }2^{g}a_{2^{one thousand}}}$ converges.

Dirichlet's test

Abel's test

Conditional and absolute convergence [edit]

For whatsoever sequence ${\displaystyle \left\{a_{ane},\ a_{2},\ a_{iii},\dots \right\}}$ , ${\displaystyle a_{north}\leq \left|a_{n}\right|}$ for all n. Therefore,

${\displaystyle \sum _{due north=one}^{\infty }a_{n}\leq \sum _{due north=one}^{\infty }\left|a_{n}\right|.}$

This ways that if ${\textstyle \sum _{n=1}^{\infty }\left|a_{n}\right|}$ converges, then ${\textstyle \sum _{northward=i}^{\infty }a_{n}}$ as well converges (but not vice versa).

If the series ${\textstyle \sum _{n=i}^{\infty }\left|a_{n}\correct|}$ converges, then the series ${\textstyle \sum _{n=one}^{\infty }a_{due north}}$ is absolutely convergent. The Maclaurin serial of the exponential function is admittedly convergent for every complex value of the variable.

If the series ${\textstyle \sum _{northward=i}^{\infty }a_{north}}$ converges but the series ${\textstyle \sum _{due north=1}^{\infty }\left|a_{n}\right|}$ diverges, and so the serial ${\textstyle \sum _{n=ane}^{\infty }a_{northward}}$ is conditionally convergent. The Maclaurin serial of the logarithm office ${\displaystyle \ln(ane+10)}$ is conditionally convergent for x = one.

The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a style that the series converges to any value, or even diverges.

Uniform convergence [edit]

Let ${\displaystyle \left\{f_{1},\ f_{2},\ f_{3},\dots \right\}}$ be a sequence of functions. The series ${\textstyle \sum _{north=1}^{\infty }f_{due north}}$ is said to converge uniformly to f if the sequence ${\displaystyle \{s_{north}\}}$ of partial sums divers by

${\displaystyle s_{northward}(ten)=\sum _{thou=1}^{n}f_{k}(x)}$

converges uniformly to f.

There is an analogue of the comparison test for infinite series of functions chosen the Weierstrass M-test.

Cauchy convergence criterion [edit]

The Cauchy convergence benchmark states that a series

${\displaystyle \sum _{n=i}^{\infty }a_{n}}$

converges if and simply if the sequence of partial sums is a Cauchy sequence. This means that for every ${\displaystyle \varepsilon >0,}$ there is a positive integer ${\displaystyle N}$ such that for all ${\displaystyle n\geq m\geq N}$ we have

${\displaystyle \left|\sum _{k=m}^{n}a_{grand}\correct|<\varepsilon ,}$

which is equivalent to

${\displaystyle \lim _{n\to \infty \atop grand\to \infty }\sum _{yard=due north}^{north+thou}a_{yard}=0.}$

See also [edit]

• Normal convergence
• List of mathematical serial

External links [edit]

• "Series", Encyclopedia of Mathematics, European monetary system Printing, 2001 [1994]
• Weisstein, Eric (2005). Riemann Serial Theorem. Retrieved May 16, 2005.

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Source: https://en.wikipedia.org/wiki/Convergent_series

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