What is the interval of convergence of #sum (x^n)/(n!) #?

Answer 1

#(-oo, oo)#

For any #x in RR#, choose #N in ZZ# such that #N > abs(x)#
#abs(sum_(n=0)^oo x^n/(n!)) = abs(sum_(n=0)^(N-1) x^n/(n!) + sum_(n=N)^oo x^n/(n!)) <= sum_(n=0)^(N-1) abs(x)^n/(n!) + sum_(n=N)^oo abs(x)^n/(n!)#
#< sum_(n=0)^(N-1) abs(x)^n/(n!) + abs(x)^N/(N!) sum_(n=0)^oo abs(x)^n/(N^n)#
The first term #sum_(n=0)^(N-1) abs(x)^n/(n!)# is a finite sum so converges.
The second term #abs(x)^N/(N!) sum_(n=0)^oo abs(x)^n/(N^n)# is the sum of a geometric series with positive common ratio #abs(x)/N < 1#, so converges.
We have shown that for any #x in (-oo,oo)#, #sum_(n=0)^oo abs(x)^n/(n!)# is bounded, that is that #sum_(n=0)^oo x^n/(n!)# is absolutely convergent. Hence it is also convergent.
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Answer 2

Utilizing the ratio test, take the absolute values.

#lim_(n->oo)abs(a_(n+1))/(abs(a_n))=lim_(n->oo) (abs(x)^(n+1)/((n+1)!))/(abs(x)^(n)/((n)!))=lim_(n->oo) absx/((n+1)!)=0#
The limit is less than 1, independent of the value of x. It follows that the series converges for all x. That is, the interval of convergence is # −∞ < x < +∞#.

In actuality, the exponential function equals the sum.

#Σ x^n/(n!)=e^x#
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Answer 3

The interval of convergence for the series ( \sum \frac{x^n}{n!} ) is from negative infinity to positive infinity, inclusive. This is because the ratio test can be applied to determine the convergence behavior of the series, and it converges for all real values of ( x ). Therefore, the interval of convergence is ( (-\infty, \infty) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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