*
*

A series is sassist to be convergent if it philosophies some limit(D"Angelo and also West 2000, p.259).

You are watching: If you delete a finite number of terms from a divergent series, does the new series still diverge?

Formally, the infinite series

*
is convergent if the sequence of partial sums


*

is convergent. Conversely, a series is divergent if the sequence of partial sums is divergent. If

*
and also
*
are convergent series, then
*
and also
*
are convergent. If
*
, then
*
and
*
both converge or both diverge. Convergence and divergence are unimpacted by deleting a finite number of terms from the start of a collection. Constant terms in the denominator of a sequence have the right to commonly be deleted without affecting convergence. All but the greatest power terms in polynomials can generally be deleted in both numerator and also denominator of a collection without affecting convergence.

If the series developed by taking the absolute values of its terms converges (in which instance it is said to be absolutely convergent), then the original series converges.

Conditions for convergence of a series have the right to be figured out in the nlinux.org Language making use of SumConvergence.

The series

*
*
*

(2)
*
*
*

(3)

both diverge by the integral test, although the latter calls for a googolplex number of terms before the partial sums exceed 10 (Zwillinger 1996, p.39). In comparison, the sums


*

(4)

(Baxley 1992; Braden 1992; Zwillinger 1996, p.39; Kreminski 1997; OEIS A115563)and


*

(5)

(OEIS A118582; Mathar 2009) converge by the integral test, although the last converges so gradually that

*
terms are necessary to obtain two-digit accuracy (Zwillinger 1996, p.39). Both can be summed making use of the Euler-Maclaurin integration formulas.


SEE ALSO: Absolute Convergence, Conditional Convergence, Convergence Tests, Convergent, Convergent Sequence, Divergent Series, Limit, Radius of Convergence, Unidevelop Convergence
REFERENCES:

Baxley, J.V. "Euler"s Constant, Taylor"s Formula, and also Slowly ConvergingSeries." Math. Mag. 65, 302-313, 1992.

Braden, B. "Calculating Sums of Infinite Series." Amer. Math. Monthly 99,649-655, 1992.

Bromwich, T.J.I"A. and MacRobert, T.M. An Summary to the Theory of Infinite Series, 3rd ed. New York: Chelsea, 1991.

D"Angelo, J.P. and West, D.B. Mathematical Thinking: Problem-Solving and Proofs, second ed. Upper Saddle River, NJ: Prentice-Hall, 2000.

Kreminski, R. "Using Simpson"s Rule to Approximate Sums of Infinite Series."College Math. J. 28, 368-376, 1997.

Mathar, R.J. "The Series Limit of

*
" />." 4 Feb 2009. http://arxiv.org/abs/0902.0789.

Sloane, N.J.A. Sequences A115563 and A118582 in "The On-Line Encyclopedia of Integer Sequences."

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and also Formulae, 30th ed. Boca Raton, FL: CRC Press, 1996.


Referenced on nlinux.org|Alpha: Convergent Series
CITE THIS AS:

Weisstein, Eric W. "Convergent Series."From nlinux.org--A nlinux.org Internet Reresource. https://mathpeople.nlinux.org.com/ConvergentSeries.html


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