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    Taylor series

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    How do I find the Taylor series for this function:

    (1/x) Integral(x to 0) (e^t - e^-t)/2t dt.

    ...up to 4 terms? And the approximate value at x = 1.

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    https://brainmass.com/math/calculus-and-analysis/example-finding-taylor-series-231041

    Solution Preview

    Solution. As e^t=1+t+t^2/2!+t^3/3!+t^4/4!+t^5/5!+...+t^n/n!+...,

    and

    e^(-t)=1-t+t^2/2!-t^3/3!+t^4/4!-t^5/5!+...+(-t)^n/n!+...,

    we have

    e^t-e^(-t)=2t+2t^3/3!+2t^5/5!+2t^7/7!+...+2t^(2n-1)/(2n-1)!+...

    Hence,

    [e^t-e^(-t)]/(2t)=1+t^2/3!+t^4/5!+t^6/7!+...+t^(2n-2)/(2n-1)!+... ...

    Solution Summary

    This provides an example of finding a Taylor series.

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