# Proof of Asymptotic Equality: $\sum_{n=0}^\infty a_n\frac{x^n}{n!} \sim ae^x$

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• Euge
In summary, asymptotic equality is a mathematical concept that describes the behavior of two functions as their inputs approach a certain value. This concept is represented by the symbol $\sim$ and is important in understanding the limiting behavior of functions and approximating complex functions. It is widely used in various fields of mathematics and science, including calculus, statistics, physics, and computer science.
Euge
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MHB
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If ##(a_n)## is a sequence of real numbers with ##\lim a_n = a##, show that $$\sum_{n = 0}^\infty a_n\frac{x^n}{n!} \sim ae^x$$ as ##x\to \infty##.

topsquark and malawi_glenn
For any ##\epsilon >0## there exist ##N## such that if ##n>N##, ##|a_n-a| < \epsilon##.

|（RHS-LHS)/RHS |＝
$$|\frac{\sum_{n=0}^\infty (a-a_n)\frac{x^n}{n!}}{ae^x}|$$$$< \frac{\sum_{n=0}^N |(a-a_n)\frac{x^n} {n!}|+\epsilon \sum_{n=N+1}^\infty \frac{x^n}{n!}}{|a|e^x}$$
$$= \frac{\sum_{n=0}^N (|a-a_n|-\epsilon)\frac{x^n} {n!}+\epsilon e^x}{|a|e^x}\rightarrow \frac{\epsilon}{|a|}$$
##\epsilon## can be taken as small as we like. So the given asymptotic equality is proved.

Last edited:
malawi_glenn, Greg Bernhardt and topsquark

## 1. What is the significance of the notation $\sim$ in the statement of asymptotic equality?

The notation $\sim$ is used to indicate that the two quantities being compared have the same asymptotic behavior. In other words, as the input variable (in this case, x) approaches infinity, the two expressions will have a similar growth rate.

## 2. How is the coefficient a in the statement of asymptotic equality determined?

The coefficient a is determined by the value of the first term in the series, a0. This term is often referred to as the leading term and will have the largest impact on the overall behavior of the series as x approaches infinity.

## 3. Can asymptotic equality be used to determine the exact value of a series?

No, asymptotic equality only provides information about the behavior of a series as the input variable approaches infinity. It does not give an exact value for the series, but rather an approximation that becomes more accurate as x gets larger.

## 4. How is the concept of asymptotic equality used in real-world applications?

Asymptotic equality is commonly used in fields such as physics, engineering, and computer science to approximate complex functions and make predictions about their behavior. It allows for simplification of calculations and can provide insight into the overall behavior of a system.

## 5. Is asymptotic equality always valid for all series?

No, asymptotic equality is only valid for certain types of series, such as power series. It may not hold true for other types of series, such as alternating series or divergent series.

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