When Is A Series Convergent

Mathematical series with a finite sum

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

${\displaystyle S=a_{0}+a_{one}+a_{2}+\cdots =\sum _{yard=0}^{\infty }a_{g}.}$

The due north thursday fractional sum Due south _{due north} is the sum of the first n terms of the sequence; that is,

${\displaystyle S_{north}=\sum _{chiliad=1}^{north}a_{m}.}$

A serial is convergent (or converges) if the sequence ${\displaystyle (S_{1},S_{two},S_{3},\dots )}$ of its partial sums tends to a limit; that means that, when adding one ${\displaystyle a_{thou}}$ later on the other in the order given by the indices, one gets fractional sums that become closer and closer to a given number. More precisely, a series converges, if in that location exists a number ${\displaystyle \ell }$ such that for every arbitrarily modest positive number ${\displaystyle \varepsilon }$ , there is a (sufficiently large) integer ${\displaystyle N}$ such that for all ${\displaystyle n\geq N}$ ,

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

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

The same note

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

is used for the serial, and, if information technology 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 every bit well as the issue of this addition, which is chosen the sum of a and b.

Any series that is not 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 series converges or diverges.

If the blueish series, ${\displaystyle \Sigma b_{due north}}$ , can be proven to converge, and then the smaller series, ${\displaystyle \Sigma a_{n}}$ must converge. Past contraposition, if the red series ${\displaystyle \Sigma a_{due north}}$ is proven to diverge, then ${\displaystyle \Sigma b_{n}}$ must also diverge.

Comparison exam. The terms of the sequence ${\displaystyle \left\{a_{n}\right\}}$ are compared to those of some other sequence ${\displaystyle \left\{b_{north}\right\}}$ . If, for all n, ${\displaystyle 0\leq \ a_{northward}\leq \ b_{n}}$ , and ${\textstyle \sum _{north=1}^{\infty }b_{n}}$ converges, and then so does ${\textstyle \sum _{n=1}^{\infty }a_{n}.}$

Even so, if, for all northward, ${\displaystyle 0\leq \ b_{n}\leq \ a_{n}}$ , and ${\textstyle \sum _{northward=i}^{\infty }b_{n}}$ diverges, so so does ${\textstyle \sum _{n=1}^{\infty }a_{n}.}$

Ratio test. Presume that for all n, ${\displaystyle a_{n}}$ is not zero. Suppose that in that location exists ${\displaystyle r}$ such that

${\displaystyle \lim _{n\to \infty }\left|{\frac {a_{north+1}}{a_{northward}}}\right|=r.}$

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

Root test or nth root test. Suppose that the terms of the sequence in question are non-negative. Define r as follows:

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

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

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

The ratio test and the root test are both based on comparison with a geometric series, and equally such they work in similar situations. In fact, if the ratio test works (significant that the limit exists and is non equal to ane) and so and so does the root test; the converse, however, is non true. The root test is therefore more by and large applicable, but as a practical matter the limit is frequently hard to compute for unremarkably seen types of serial.

Integral exam. The series can be compared to an integral to establish convergence or deviation. Allow ${\displaystyle f(northward)=a_{north}}$ be a positive and monotonically decreasing office. If

${\displaystyle \int _{ane}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty ,}$

and so the series converges. But if the integral diverges, so the series does so as well.

Limit comparing examination. If ${\displaystyle \left\{a_{n}\correct\},\left\{b_{n}\right\}>0}$ , and the limit ${\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{north}}}}$ exists and is non zero, so ${\textstyle \sum _{n=one}^{\infty }a_{n}}$ converges if and only if ${\textstyle \sum _{n=ane}^{\infty }b_{n}}$ converges.

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

Cauchy condensation examination. If ${\displaystyle \left\{a_{n}\right\}}$ is a positive monotone decreasing sequence, then ${\textstyle \sum _{n=1}^{\infty }a_{north}}$ converges if and only if ${\textstyle \sum _{k=1}^{\infty }2^{k}a_{2^{chiliad}}}$ converges.

Dirichlet's exam

Abel's examination

Conditional and accented convergence [edit]

For whatever sequence ${\displaystyle \left\{a_{1},\ a_{two},\ a_{3},\dots \right\}}$ , ${\displaystyle a_{n}\leq \left|a_{n}\right|}$ for all n. Therefore,

${\displaystyle \sum _{n=one}^{\infty }a_{n}\leq \sum _{due north=1}^{\infty }\left|a_{n}\right|.}$

This means that if ${\textstyle \sum _{n=ane}^{\infty }\left|a_{north}\right|}$ converges, and so ${\textstyle \sum _{due north=1}^{\infty }a_{northward}}$ also converges (but not vice versa).

If the serial ${\textstyle \sum _{northward=ane}^{\infty }\left|a_{n}\right|}$ converges, then the series ${\textstyle \sum _{north=1}^{\infty }a_{n}}$ is absolutely convergent. The Maclaurin series of the exponential part is absolutely convergent for every complex value of the variable.

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

The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the serial converges to whatsoever value, or fifty-fifty diverges.

Uniform convergence [edit]

Let ${\displaystyle \left\{f_{ane},\ f_{2},\ f_{three},\dots \right\}}$ exist a sequence of functions. The series ${\textstyle \sum _{n=1}^{\infty }f_{n}}$ is said to converge uniformly to f if the sequence ${\displaystyle \{s_{n}\}}$ of fractional sums defined by

${\displaystyle s_{n}(x)=\sum _{k=1}^{north}f_{one thousand}(x)}$

converges uniformly to f.

In that location is an counterpart of the comparison exam for infinite series of functions called the Weierstrass M-test.

Cauchy convergence criterion [edit]

The Cauchy convergence criterion states that a serial

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

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

${\displaystyle \left|\sum _{k=m}^{north}a_{k}\right|<\varepsilon ,}$

which is equivalent to

${\displaystyle \lim _{n\to \infty \atop thousand\to \infty }\sum _{one thousand=north}^{n+m}a_{yard}=0.}$

See also [edit]

• Normal convergence
• List of mathematical serial

External links [edit]

• "Serial", Encyclopedia of Mathematics, Ems Press, 2001 [1994]
• Weisstein, Eric (2005). Riemann Series Theorem. Retrieved May 16, 2005.

When Is A Series Convergent,

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

Posted by: colemanallse1994.blogspot.com

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