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Integrating sine and cosine Maclaurin remainders
Published online by Cambridge University Press: 16 February 2023
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In order to state our primary results, we must first establish some
notation. Let S-1 (x) = sin x and C-1 (x) = cos x, then for each non-negative integer n, let
these are the remainders of the Maclaurin series for sine and cosine, respectively. Note that 
for each 
and 
for each 
. It is known that
See [1] for several different proofs of the well-known fact that
the values of αn and βn for 
then follow rather easily using induction and integration by parts. (Details are provided in the Appendix.)
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- © The Authors, 2023. Published by Cambridge University Press on behalf of The Mathematical Association
References
Jameson, G. J. O., Sine, cosine and exponential integrals, Math. Gaz. 99 (July 2015) pp. 276-289.CrossRefGoogle Scholar
Stewart, S. M., Some improper integrals involving the square of the tail of the sine and cosine functions, J. Class. Anal. 16(2) (2020) pp. 91–99.Google Scholar
Gordon, R. A., Integrating the tails of two Maclaurin series, J. Class. Anal. 18(1) (2021) pp. 83–95.CrossRefGoogle Scholar
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