Approximating sinh through taylor's series

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In summary, Taylor's series is a mathematical concept used to approximate functions using polynomial expressions. It is named after mathematician Brook Taylor and is commonly used in calculus and other areas of mathematics. To approximate sinh (hyperbolic sine) through Taylor's series, we use a formula that becomes more accurate as more terms are included. The purpose of approximating sinh through this method is to simplify complex functions and get a good estimate of their value. The benefit of using Taylor's series for this purpose is that it allows us to calculate the value of sinh for any input and gives us an idea of the error in our approximation. However, there are limitations to using this method, such as the series not converging for all values of x and becoming
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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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  • #2
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?
 

1. What is Taylor's series?

Taylor's series is a mathematical concept that allows us to approximate a function using a polynomial expression. It is named after the mathematician Brook Taylor and is often used in calculus and other areas of mathematics.

2. How is sinh approximated through Taylor's series?

To approximate sinh (hyperbolic sine) through Taylor's series, we use the following formula: sinh(x) = x + (x^3)/3! + (x^5)/5! + (x^7)/7! + ... The more terms we include in the series, the more accurate our approximation will be.

3. What is the purpose of approximating sinh through Taylor's series?

The purpose of approximating sinh through Taylor's series is to find a simpler expression for a function that may be difficult to evaluate directly. It allows us to break down a complex function into smaller, manageable parts and get a good estimate of its value.

4. What is the benefit of using Taylor's series to approximate sinh?

The benefit of using Taylor's series to approximate sinh is that it allows us to calculate the value of sinh for any given input, even if we do not have a specific formula for it. It also gives us an idea of the error in our approximation and allows us to improve it by including more terms in the series.

5. Are there any limitations to approximating sinh through Taylor's series?

Yes, there are some limitations to approximating sinh through Taylor's series. One limitation is that the series may not converge for all values of x, so we need to be careful when choosing the input. Another limitation is that the more terms we include in the series, the more complex the calculations become, making it difficult to use for large inputs.

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