Approximating sinh through taylor's series

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The forum discussion focuses on approximating the hyperbolic sine function, sinh(3391014490), using Taylor's series. The user seeks guidance on determining the number of terms required to achieve an error margin within 10^-10. It is established that for large values of x, sinh(x) can be approximated by (e^x)/2. The discussion highlights that sinh(3391014490) will yield approximately 3391014490 x log_10(e) decimal digits, significantly exceeding 10^9 digits, raising concerns about the feasibility of calculating this to just 10 decimal places.

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jakey
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Hi guys,

I want to approximate sinh(3391014490) using taylor's series as i don't know of a way to compute this in c++ (Even NTL can't handle this much).

How could I compute the number of terms I need to sum so that the error is within 10^-10?

Any suggestions? thanks!
 
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sinh x = (e^x - e^-x)/2

So for x >> 1

sinh x is approximately (e^x)/2

sinh(3391014490) will have about 3391014490 x log_10(e) decimal digits
which is more than 10^9 digits. Are you sure you want to calculate this to 10 decimal places?
 

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