Calculus Examples

$\sum_{k = 1}^{\infty} k e^{k^{2}}$

Step 1

Check if the function is continuous over the summation bounds.

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Step 1.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${k | k \in ℝ}$

Step 1.2

$f (k)$ is continuous on $[1, \infty)$ .

$The function is continuous.$

Step 2

Check if the function is positive over the bounds.

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Step 2.1

Set up an inequality.

$k e^{k^{2}} > 0$

Step 2.2

Solve the inequality.

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Step 2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$k = 0$

$e^{k^{2}} = 0$

Step 2.2.2

Set $k$ equal to $0$ .

$k = 0$

Step 2.2.3

Set $e^{k^{2}}$ equal to $0$ and solve for $k$ .

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Step 2.2.3.1

Set $e^{k^{2}}$ equal to $0$ .

$e^{k^{2}} = 0$

Step 2.2.3.2

Solve $e^{k^{2}} = 0$ for $k$ .

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Step 2.2.3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{k^{2}}) = \ln (0)$

Step 2.2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.2.3.2.3

There is no solution for $e^{k^{2}} = 0$

No solution

Step 2.2.4

The final solution is all the values that make $k e^{k^{2}} > 0$ true.

$k = 0$

Step 2.2.5

The solution consists of all of the true intervals.

$k > 0$

Step 3

Determine where the function is decreasing.

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Step 3.1

Write $k e^{k^{2}}$ as a function.

$f (k) = k e^{k^{2}}$

Step 3.2

Find the first derivative.

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Step 3.2.1

Find the first derivative.

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Step 3.2.1.1

Differentiate using the Product Rule which states that $\frac{d}{d k} [f (k) g (k)]$ is $f (k) \frac{d}{d k} [g (k)] + g (k) \frac{d}{d k} [f (k)]$ where $f (k) = k$ and $g (k) = e^{k^{2}}$ .

$k \frac{d}{d k} [e^{k^{2}}] + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d k} [f (g (k))]$ is $f' (g (k)) g' (k)$ where $f (k) = e^{k}$ and $g (k) = k^{2}$ .

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Step 3.2.1.2.1

To apply the Chain Rule, set $u$ as $k^{2}$ .

$k (\frac{d}{d u} [e^{u}] \frac{d}{d k} [k^{2}]) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$k (e^{u} \frac{d}{d k} [k^{2}]) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.2.3

Replace all occurrences of $u$ with $k^{2}$ .

$k (e^{k^{2}} \frac{d}{d k} [k^{2}]) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d k} [k^{n}]$ is $n k^{n - 1}$ where $n = 2$ .

$k (e^{k^{2}} (2 k)) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.4

Raise $k$ to the power of $1$ .

$k^{1} k (e^{k^{2}} \cdot (2)) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.5

Raise $k$ to the power of $1$ .

$k^{1} k^{1} (e^{k^{2}} \cdot (2)) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$k^{1 + 1} (e^{k^{2}} \cdot (2)) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.7

Simplify the expression.

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Step 3.2.1.7.1

Add $1$ and $1$ .

$k^{2} (e^{k^{2}} \cdot (2)) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.7.2

Move $2$ to the left of $e^{k^{2}}$ .

$k^{2} (2 \cdot e^{k^{2}}) + e^{k^{2}} \frac{d}{d k} [k]$

Step 3.2.1.8

Differentiate using the Power Rule which states that $\frac{d}{d k} [k^{n}]$ is $n k^{n - 1}$ where $n = 1$ .

$k^{2} (2 e^{k^{2}}) + e^{k^{2}} \cdot 1$

Step 3.2.1.9

Multiply $e^{k^{2}}$ by $1$ .

$k^{2} (2 e^{k^{2}}) + e^{k^{2}}$

Step 3.2.1.10

Simplify.

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Step 3.2.1.10.1

Reorder terms.

$2 e^{k^{2}} k^{2} + e^{k^{2}}$

Step 3.2.1.10.2

Reorder factors in $2 e^{k^{2}} k^{2} + e^{k^{2}}$ .

$f' (k) = 2 k^{2} e^{k^{2}} + e^{k^{2}}$

Step 3.2.2

The first derivative of $f (k)$ with respect to $k$ is $2 k^{2} e^{k^{2}} + e^{k^{2}}$ .

$2 k^{2} e^{k^{2}} + e^{k^{2}}$

Step 3.3

Set the first derivative equal to $0$ then solve the equation $2 k^{2} e^{k^{2}} + e^{k^{2}} = 0$ .

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Step 3.3.1

Set the first derivative equal to $0$ .

$2 k^{2} e^{k^{2}} + e^{k^{2}} = 0$

Step 3.3.2

Factor $e^{k^{2}}$ out of $2 k^{2} e^{k^{2}} + e^{k^{2}}$ .

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Step 3.3.2.1

Factor $e^{k^{2}}$ out of $2 k^{2} e^{k^{2}}$ .

$e^{k^{2}} (2 k^{2}) + e^{k^{2}} = 0$

Step 3.3.2.2

Multiply by $1$ .

$e^{k^{2}} (2 k^{2}) + e^{k^{2}} \cdot 1 = 0$

Step 3.3.2.3

Factor $e^{k^{2}}$ out of $e^{k^{2}} (2 k^{2}) + e^{k^{2}} \cdot 1$ .

$e^{k^{2}} (2 k^{2} + 1) = 0$

Step 3.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{k^{2}} = 0$

$2 k^{2} + 1 = 0$

Step 3.3.4

Set $e^{k^{2}}$ equal to $0$ and solve for $k$ .

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Step 3.3.4.1

Set $e^{k^{2}}$ equal to $0$ .

$e^{k^{2}} = 0$

Step 3.3.4.2

Solve $e^{k^{2}} = 0$ for $k$ .

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Step 3.3.4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{k^{2}}) = \ln (0)$

Step 3.3.4.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 3.3.4.2.3

There is no solution for $e^{k^{2}} = 0$

No solution

Step 3.3.5

Set $2 k^{2} + 1$ equal to $0$ and solve for $k$ .

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Step 3.3.5.1

Set $2 k^{2} + 1$ equal to $0$ .

$2 k^{2} + 1 = 0$

Step 3.3.5.2

Solve $2 k^{2} + 1 = 0$ for $k$ .

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Step 3.3.5.2.1

Subtract $1$ from both sides of the equation.

$2 k^{2} = - 1$

Step 3.3.5.2.2

Divide each term in $2 k^{2} = - 1$ by $2$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $2 k^{2} = - 1$ by $2$ .

$\frac{2 k^{2}}{2} = \frac{- 1}{2}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 k^{2}}{2} = \frac{- 1}{2}$

Step 3.3.5.2.2.2.1.2

Divide $k^{2}$ by $1$ .

$k^{2} = \frac{- 1}{2}$

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Move the negative in front of the fraction.

$k^{2} = - \frac{1}{2}$

Step 3.3.5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$k = \pm \sqrt{- \frac{1}{2}}$

Step 3.3.5.2.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 3.3.5.2.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 3.3.5.2.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$k = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 3.3.5.2.4.1.2

Rewrite $1$ as $1^{2}$ .

$k = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 3.3.5.2.4.2

Pull terms out from under the radical.

$k = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 3.3.5.2.4.3

One to any power is one.

$k = \pm i \sqrt{\frac{1}{2}}$

Step 3.3.5.2.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$k = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 3.3.5.2.4.5

Any root of $1$ is $1$ .

$k = \pm i \frac{1}{\sqrt{2}}$

Step 3.3.5.2.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$k = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 3.3.5.2.4.7

Combine and simplify the denominator.

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Step 3.3.5.2.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$k = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.3.5.2.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$k = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 3.3.5.2.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$k = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 3.3.5.2.4.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$k = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.3.5.2.4.7.5

Add $1$ and $1$ .

$k = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 3.3.5.2.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.3.5.2.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$k = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 3.3.5.2.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k = \pm i \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 3.3.5.2.4.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$k = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.5.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.3.5.2.4.7.6.4.1

Cancel the common factor.

$k = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 3.3.5.2.4.7.6.4.2

Rewrite the expression.

$k = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 3.3.5.2.4.7.6.5

Evaluate the exponent.

$k = \pm i \frac{\sqrt{2}}{2}$

Step 3.3.5.2.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$k = \pm \frac{i \sqrt{2}}{2}$

Step 3.3.5.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.5.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$k = \frac{i \sqrt{2}}{2}$

Step 3.3.5.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$k = - \frac{i \sqrt{2}}{2}$

Step 3.3.5.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$k = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 3.3.6

The final solution is all the values that make $e^{k^{2}} (2 k^{2} + 1) = 0$ true.

$k = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 3.4

There are no values of $k$ in the domain of the original problem where the derivative is $0$ or undefined.

No critical points found

Step 3.5

No points make the derivative $f' (k) = 2 k^{2} e^{k^{2}} + e^{k^{2}}$ equal to $0$ or undefined. The interval to check if $f (k) = k e^{k^{2}}$ is increasing or decreasing is $(- \infty, \infty)$ .

$(- \infty, \infty)$

Step 3.6

Substitute any number, such as $1$ , from the interval $(- \infty, \infty)$ in the derivative $f' (k) = 2 k^{2} e^{k^{2}} + e^{k^{2}}$ to check if the result is negative or positive. If the result is negative, the graph is decreasing on the interval $(- \infty, \infty)$ . If the result is positive, the graph is increasing on the interval $(- \infty, \infty)$ .

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Step 3.6.1

Replace the variable $k$ with $1$ in the expression.

$f' (1) = 2 {(1)}^{2} e^{{(1)}^{2}} + e^{{(1)}^{2}}$

Step 3.6.2

Simplify the result.

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Step 3.6.2.1

Simplify each term.

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Step 3.6.2.1.1

One to any power is one.

$f' (1) = 2 \cdot (1 e^{{(1)}^{2}}) + e^{{(1)}^{2}}$

Step 3.6.2.1.2

Multiply $2$ by $1$ .

$f' (1) = 2 e^{{(1)}^{2}} + e^{{(1)}^{2}}$

Step 3.6.2.1.3

One to any power is one.

$f' (1) = 2 e + e^{{(1)}^{2}}$

Step 3.6.2.1.4

Simplify.

$f' (1) = 2 e + e^{{(1)}^{2}}$

Step 3.6.2.1.5

One to any power is one.

$f' (1) = 2 e + e$

Step 3.6.2.1.6

Simplify.

$f' (1) = 2 e + e$

Step 3.6.2.2

Add $2 e$ and $e$ .

$f' (1) = 3 e$

Step 3.6.2.3

The final answer is $3 e$ .

$3 e$

Step 3.7

The result of substituting $1$ into $f' (k) = 2 k^{2} e^{k^{2}} + e^{k^{2}}$ is $3 e$ , which is positive, so the graph is increasing on the interval $(- \infty, \infty)$ .

Increasing on $(- \infty, \infty)$ since $2 k^{2} e^{k^{2}} + e^{k^{2}} > 0$

Step 3.8

Increasing over the interval $(- \infty, \infty)$ means that the function is always increasing.

$Always Increasing$

Step 4

The integral test does not apply because the function is not always decreasing from $1$ to $\infty$ .

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