2 years ago

The sum of a number and its square is 2. Find the number(s).

1 answer:

6 0

Let x represent the number.
.. x^2 +x = 2
.. x^2 +x +1/4 = 9/4 . . . . add 1/4 to complete the square
.. x +1/2 = ±3/2 . . . . . . . .square root
.. x = -2, 1

Numbers meeting that requirement are -2 and 1.

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What is the sum of the series 1 + ln3 + (ln3)^2/2!?

tresset_1 [31]

$\displaystyle\sum_{n\ge0}\frac{(\ln3)^n}{n!}=e^{\ln3}=3$

which follows from the power series representation for $e^x$ ,

$e^x=\displaystyle\sum_{n\ge0}\frac{x^n}{n!}$

and taking $x=\ln3$ .