# A note on the summation of slowly convergent alternating series

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## Abstract

The*k*th term of the infinite series\(\sum\nolimits_{k = 0}^\infty {( - 1)^k [\ln (k + 2)]^{ - 0.1} }\) is larger than 0.5 whenever*k* <*k*_{0}, where*k*_{0} + 1 =*e*^{1024}. To sum this series correct to order 10^{−1} using direct summation seems an impossible task, notwithstanding the power of modern computers. This note will present an alternative approach to those classical methods (the Euler transformation is one) which can accurately sum such series. The theory to be presented has the added advantage of providing accurate bounds for the error in the approximate result. The method used will be Euler-Maclaurin summation, revitalised by computer algebra.

## Key words

Summation slowly convergent alternating series Euler-Maclaurin formula## Preview

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© BIT Foundation 1996