Sum of (n+1) terms in exponential series

[ssd]

Homework Statement

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Homework Equations

None

The Attempt at a Solution

I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

 

[fresh_42]

Homework Statement

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Homework Equations

None

The Attempt at a Solution

I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

As far as I know, all you can get is $e^x -\sum_{k=0}^n \frac{x^k}{k!} = r_n(x)$ with some boundaries $c \le r_n(x) \le C$

 

[Ray Vickson]

Homework Statement

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Homework Equations

None

The Attempt at a Solution

I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

According to Maple, the sum can be expressed in terms of an incomplete Gamma function $\Gamma(n+1,x)$ and some other factors, but I am not sure you would call that "simple".

 

[Mark44]

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Just to check, S isn't given like this, is it?
S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n! + ...

 

[ssd]

Just to check, S isn't given like this, is it?
S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n! + ...

No, only n+1 terms.

 

[LCKurtz]

Homework Statement

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Those already look like simple terms to me.

 

[coolul007]

Homework Statement

S = 1+ x/1! +x²/2! +x³/3! +...+xⁿ/n!
To find S in simple terms.

Homework Equations

None

The Attempt at a Solution

I tried with Taylor's expansion, coshx and sinhx expansions. But cannot see consequence.

s=e^x

 

[Ray Vickson]

s=e^x

No: absolutely not! The infinite sum is $e^x$ but--at least in the initial post--the OP is asking about the finite sum, just for the first $n+1$ terms of the exponential series.