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Question: The expansion of \( {e^x} \) is (A) \( \sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{r}} \) (B) \...

The expansion of ex{e^x} is
(A) ∑r=0∞xrr\sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{r}}
(B) ∑r=0∞xrr!\sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{{r!}}}
(C) ∑r=0∞xr+1r+1\sum\limits_{r = 0}^\infty {\dfrac{{{x^{r + 1}}}}{{r + 1}}}
(D) ∑r=0∞xr+1(r+1)!\sum\limits_{r = 0}^\infty {\dfrac{{{x^{r + 1}}}}{{(r + 1)!}}}

Explanation

Solution

Hint : Write down the expansion formula of ex{e^x} . Then expand the summation of all the options. Then check which option matches the expansion formula of ex{e^x} . To expand the summation sign. First put the lower value i.e. r=0r = 0 in the given expression. Then write the sign of addition. Then put r=1r = 1 in the given expression. Then put the sign of addition. And so on. . .

Complete step-by-step answer :
We know that, expansion formula of ex{e^x} is given by
⇒ex=1+x1!+x22!+x33!+....\Rightarrow {e^x} = 1 + \dfrac{x}{{1!}} + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + .... . . . (1)
Now, let us expand the options
Option (A)
∑r=0∞xrr\sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{r}}
Clearly expression in option (A) does not exist because when you put r=0r = 0 . Then the denominator will be equal to zero and hence the fraction will be ∞\infty .
So, we cannot expand it.
Therefore, option (A) is incorrect.
Option (B)
∑r=0∞xrr!=x00!+x11!+x22!+x33!+...\sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{{r!}}} = \dfrac{{{x^0}}}{{0!}} + \dfrac{{{x^1}}}{{1!}} + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + ...
Now, we know that, 0!=10! = 1 and x0=1{x^0} = 1 . Therefore, we can simplify the above expression as
=1+x11!+x22!+x33!+...= 1 + \dfrac{{{x^1}}}{{1!}} + \dfrac{{{x^2}}}{{2!}} + \dfrac{{{x^3}}}{{3!}} + ... . . . (2)
We can observe that, expansion of equation (1) and equation (2) is equal.
Thus, ex=∑r=0∞xrr!{e^x} = \sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{{r!}}}
Therefore, from the above explanation, the correct answer is, option (B) ∑r=0∞xrr!\sum\limits_{r = 0}^\infty {\dfrac{{{x^r}}}{{r!}}}
So, the correct answer is “Option B”.

Note : To solve this question, you need to know the expansion formula of ex{e^x} . Because this is a memory based question. If you don’t know the expansion of ex{e^x} , then you cannot solve it.
This question can also be solved smartly without pen and paper. If you observe the options closely, then only option (B) gives the first term equal to 11 . Which is the first term of the expansion of ex{e^x} . Hence, only option (B) can be the correct answer.