QuestionIdentify the incorrect expression for the sum $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ from the options given.

Studdy Solution

STEP 1

Assumptions1. The given sum is $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ . We need to find the incorrect expression among the given options3. The factorial function $r!$ is defined as the product of all positive integers less than or equal to $r$

STEP 2

First, let's write out the given sum by substituting the values of $r$ from $0$ to $$.
$-x \sum_{r=0}^{} \frac{x^{r+1}}{r !} = -x\left[\frac{x^{0+1}}{0 !}+\frac{x^{1+1}}{1 !}+\frac{x^{2+1}}{2 !}+\frac{x^{+1}}{ !}\right]$

STEP 3

Now, let's simplify the expression by calculating the factorial values and simplifying the powers of $x$ .
$-x\left[\frac{x^{0+1}}{0 !}+\frac{x^{1+1}}{1 !}+\frac{x^{2+1}}{2 !}+\frac{x^{3+1}}{3 !}\right] = -x\left[\frac{x}{1}+\frac{x^{2}}{1}+\frac{x^{3}}{2}+\frac{x^{}}{6}\right]$

STEP 4

Now, let's compare this expression with the given options.Option a is the same as our expression, so it is correct.
Option d is also the same as our expression, so it is correct.

STEP 5

For option b, let's simplify it $\frac{-x \cdot x^{1}}{0 !}+\frac{x \cdot x^{2}}{1 !}+\frac{x \cdot x^{3}}{2 !}+\frac{x \cdot x^{4}}{3 !} = -x^{2} + x^{3} + \frac{x^{4}}{2} + \frac{x^{5}}{}$ This is not the same as our expression, so option b is incorrect.

STEP 6

For option c, let's simplify it $\frac{-6 x^{2}-6 x^{3}-3 x^{4}-x^{5}}{6} = -x^{2} - x^{3} - \frac{x^{4}}{2} - \frac{x^{5}}{6}$ This is not the same as our expression, so option c is also incorrect.
However, the question asks for the expression that is NOT correct, so the answer is option b.

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