How Accurate Are Partial Sums in Estimating e^N?

In summary, the purpose of including error terms in Taylor Series is to account for the difference between the exact value of a function and its approximation using a finite number of terms. These error terms are calculated using the remainder term formula and can be negative. They affect the convergence of the series and can be used to estimate the error in the approximation. By adding more terms, the error decreases and the approximation becomes more accurate.
  • #1
Nurdan
4
0
It is known that
[tex] \sum\limits_{k = 0}^\infty {\frac{{N^k }}{{k!}}} = e^N[/tex]


I am looking for any asymptotic approximation which gives

[tex] \sum\limits_{k = 0}^M {\frac{{N^k }}{{k!}}} = ? [/tex]
where [tex]M\leq N[/tex] an integer.


This is not an homework
 
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  • #2
I'm only being a little facetious if I point out that the sum is asymptotically equal to e^N.

Want more accuracy? It's better approximated by e^N-[x^(N+1)]/(N+1)!
 

Related to How Accurate Are Partial Sums in Estimating e^N?

1. What is the purpose of including error terms in Taylor Series?

The purpose of including error terms in Taylor Series is to account for the difference between the exact value of a function and its approximation using a finite number of terms. These error terms give us an idea of how accurate our approximation is and help us improve the accuracy by adding more terms to the series.

2. How are error terms calculated in Taylor Series?

Error terms in Taylor Series are calculated using the remainder term formula: Rn(x) = f(n+1)(c)(x-a)^n+1 / (n+1)!, where f(n+1)(c) represents the value of the (n+1)th derivative of the function at some point c between the original point a and the point at which the approximation is being made.

3. Can error terms be negative in Taylor Series?

Yes, error terms can be negative in Taylor Series. The error term represents the difference between the exact value and the approximation, so it can be positive or negative depending on the direction of the difference.

4. How do error terms affect the convergence of Taylor Series?

The error terms in Taylor Series affect the convergence of the series by determining how accurate the approximation is. As we add more terms to the series, the error terms decrease and the approximation becomes more accurate. If the error terms approach 0, then the series will converge and accurately represent the function.

5. Can error terms be used to estimate the error in Taylor Series?

Yes, error terms can be used to estimate the error in Taylor Series. By calculating the error term using the remainder term formula, we can get an idea of how large the error is in our approximation. This can help us determine how many terms we need to add to the series to get a more accurate approximation.

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