How come the sum of the reciprocals of the factorials of the integers from zero to infinity equal e = 2.718...

Question

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Please explain/give a website^_^
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Answer 1

Using Taylor's theorem, one can prove that

e^x=\sum_{k=0}^{infinity} x^k/k!

for all values of x.

Hence,

e=e^1=sum_{k=0}^{infinity} 1^k/k!=
sum_{k=0}^{infinity} 1/k!

Answer 2

U cud actually try it ur self..

after a while, when u start getting the 2.718 mark, the number you are adding gets very close to zero that u are actually adding a very small amount that it does not affect the answer that much..

Answer 3

To elaborate on Jim's answer:
If y=sum x^k /k!
where k=0 to infty,
then taking the derivative term by term, you can see that
dy/dx=y.
This means that y=Ce^x for some constant C. By plugging in x=0, we see C=1, so
e^x =sum x^k /k!
Also, the radius of convergence of this series is infinite, so you can plug in x=1 to find
e=sum 1/k!

Answer 4

there are number of ways to prove this .

Answer 5

If you don't believe it - work it out for yourself.

Answer 6

say what??????