Which expressions describe $\frac{e}{2}-\frac{e^2}{4}+\frac{e^3}{6}-\frac{e^4}{8}+....$

Question

I'm taking single variable Calculus on Coursera and asked following :

Which of the following expressions describes the sum $\frac{e}{2}-\frac{e^2}{4}+\frac{e^3}{6}-\frac{e^4}{8}+....$ :
$$1. \sum_{k=0}^\infty (-1)^{k+1}\frac{e^k}{2k} $$ $$2. \sum_{k=0}^\infty (-1)^{k}\frac{e^{k+1}}{2k+2} $$ $$3. \sum_{k=0}^\infty (-1)^{k+1}\frac{e^{k+1}}{2k+2} $$ $$4. \sum_{k=1}^\infty (-1)^{k}\frac{e^{k+1}}{2k+2} $$ $$5. \sum_{k=1}^\infty (-1)^{k+1}\frac{e^k}{2k} $$ $$6. \sum_{k=1}^\infty (-1)^{k}\frac{e^k}{2k} $$

Question screenshot :

My solution :

$$ e^x = \sum_{k=0}^\infty\frac{x^k}{k!}$$

Using k = 1 for $\sum_{k=0}^\infty (-1)^{k+1}\frac{e^k}{2k} $

$$(-1)^{1+1}\frac{e^1}{2} $$

=

$$1\frac{e^1}{2} $$

=

$$\frac{e^1}{2} $$

Using k = 2 for $\sum_{k=0}^\infty (-1)^{k+1}\frac{e^k}{2k} $

$$(-1)^{2+1}\frac{e^2}{4} $$

=

$$-1\frac{e^2}{4} $$

=

$$-\frac{e^2}{4} $$

$\sum_{k=0}^\infty (-1)^{k+1}\frac{e^k}{2k} $ matches $k=1, k=2$ the sum expression $\frac{e}{2}-\frac{e^2}{4}$ . Is the solution to keep plugging in values for k and checking which match the sum expressions ? But this does not tell me $\frac{e}{2}-\frac{e^2}{4}+\frac{e^3}{6}-\frac{e^4}{8}+....$ as it just provides an answer for first 4 k terms.

Is there a more intuitive method to solve this ?

Answer 1

Notice that $$\frac{e}{2}-\frac{e^2}{4}+\frac{e^3}{6}-\dots = \sum_{k=1}^{\infty} (-1)^{k-1}\frac{e^{k}}{2k}=\sum_{k=0}^{\infty}(-1)^{k}\frac{e^{k+1}}{2(k+1)}.$$ The idea is to simply write it using the sum notation and then do a substitution to get the correct summation.

So I would definitely answer $4$ (I guess there is a mistake in the exponents as written down in the question).

Answer 2

Obviously, in all the given series, the signs go alternating, the exponents increase by one and the denominators increase by two from term to term. Then if the first term is right, so will all the following ones.

• $-\dfrac10$;

• $\color{green}{+\dfrac e2}$;

• $-\dfrac e2$;

• $-\dfrac{e^2}4$;

• $\color{green}{+\dfrac e2}$;

• $-\dfrac e2$.

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If you are interested by the value of the sum, the answer is easy: it doesn't exist, as the terms grow to infinity.