Calculus Examples

$\sum_{n = 4}^{\infty} \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$\frac{1}{\sqrt{n}} \cdot \frac{\sqrt{n}}{\sqrt{n}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.1

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

$\frac{\sqrt{n}}{\sqrt{n} \sqrt{n}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.2

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

$\frac{\sqrt{n}}{{\sqrt{n}}^{1} \sqrt{n}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.3

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

$\frac{\sqrt{n}}{{\sqrt{n}}^{1} {\sqrt{n}}^{1}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.4

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

$\frac{\sqrt{n}}{{\sqrt{n}}^{1 + 1}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.5

Add $1$ and $1$ .

$\frac{\sqrt{n}}{{\sqrt{n}}^{2}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

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

Step 1.2.6.2

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

$\frac{\sqrt{n}}{n^{\frac{1}{2} \cdot 2}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.6.3

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

$\frac{\sqrt{n}}{n^{\frac{2}{2}}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.4.1

Cancel the common factor.

$\frac{\sqrt{n}}{n^{\frac{2}{2}}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{\sqrt{n}}{n^{1}} - \frac{1}{\sqrt{n + 1}}$

Step 1.2.6.5

Simplify.

$\frac{\sqrt{n}}{n} - \frac{1}{\sqrt{n + 1}}$

Step 1.3

Multiply $\frac{1}{\sqrt{n + 1}}$ by $\frac{\sqrt{n + 1}}{\sqrt{n + 1}}$ .

$\frac{\sqrt{n}}{n} - (\frac{1}{\sqrt{n + 1}} \cdot \frac{\sqrt{n + 1}}{\sqrt{n + 1}})$

Step 1.4

Combine and simplify the denominator.

Tap for more steps...

Step 1.4.1

Multiply $\frac{1}{\sqrt{n + 1}}$ by $\frac{\sqrt{n + 1}}{\sqrt{n + 1}}$ .

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{\sqrt{n + 1} \sqrt{n + 1}}$

Step 1.4.2

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

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{\sqrt{n + 1}}^{1} \sqrt{n + 1}}$

Step 1.4.3

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

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{\sqrt{n + 1}}^{1} {\sqrt{n + 1}}^{1}}$

Step 1.4.4

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

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{\sqrt{n + 1}}^{1 + 1}}$

Step 1.4.5

Add $1$ and $1$ .

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{\sqrt{n + 1}}^{2}}$

Step 1.4.6

Rewrite ${\sqrt{n + 1}}^{2}$ as $n + 1$ .

Tap for more steps...

Step 1.4.6.1

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

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

Step 1.4.6.2

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

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{(n + 1)}^{\frac{1}{2} \cdot 2}}$

Step 1.4.6.3

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

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

Step 1.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.6.4.1

Cancel the common factor.

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

Step 1.4.6.4.2

Rewrite the expression.

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{{(n + 1)}^{1}}$

Step 1.4.6.5

Simplify.

$\frac{\sqrt{n}}{n} - \frac{\sqrt{n + 1}}{n + 1}$

Step 2

To write $\frac{\sqrt{n}}{n}$ as a fraction with a common denominator, multiply by $\frac{n + 1}{n + 1}$ .

$\frac{\sqrt{n}}{n} \cdot \frac{n + 1}{n + 1} - \frac{\sqrt{n + 1}}{n + 1}$

Step 3

To write $- \frac{\sqrt{n + 1}}{n + 1}$ as a fraction with a common denominator, multiply by $\frac{n}{n}$ .

$\frac{\sqrt{n}}{n} \cdot \frac{n + 1}{n + 1} - \frac{\sqrt{n + 1}}{n + 1} \cdot \frac{n}{n}$

Step 4

Write each expression with a common denominator of $n (n + 1)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{\sqrt{n}}{n}$ by $\frac{n + 1}{n + 1}$ .

$\frac{\sqrt{n} (n + 1)}{n (n + 1)} - \frac{\sqrt{n + 1}}{n + 1} \cdot \frac{n}{n}$

Step 4.2

Multiply $\frac{\sqrt{n + 1}}{n + 1}$ by $\frac{n}{n}$ .

$\frac{\sqrt{n} (n + 1)}{n (n + 1)} - \frac{\sqrt{n + 1} n}{(n + 1) n}$

Step 4.3

Reorder the factors of $(n + 1) n$ .

$\frac{\sqrt{n} (n + 1)}{n (n + 1)} - \frac{\sqrt{n + 1} n}{n (n + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{\sqrt{n} (n + 1) - \sqrt{n + 1} n}{n (n + 1)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $n^{\frac{1}{2}}$ by $n$ by adding the exponents.

Tap for more steps...

Step 6.4.1

Multiply $n^{\frac{1}{2}}$ by $n$ .

Tap for more steps...

Step 6.4.1.1

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

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

Step 6.4.1.2

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

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

Step 6.4.2

Write $1$ as a fraction with a common denominator.

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

Step 6.4.3

Combine the numerators over the common denominator.

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

Step 6.4.4

Add $1$ and $2$ .

$\frac{n^{\frac{3}{2}} + n^{\frac{1}{2}} \cdot 1 - {(n + 1)}^{\frac{1}{2}} n}{n (n + 1)}$

Step 6.5

Multiply $n^{\frac{1}{2}}$ by $1$ .

$\frac{n^{\frac{3}{2}} + n^{\frac{1}{2}} - {(n + 1)}^{\frac{1}{2}} n}{n (n + 1)}$

Step 6.6

Reorder terms.

$\frac{- n {(n + 1)}^{\frac{1}{2}} + n^{\frac{3}{2}} + n^{\frac{1}{2}}}{n (n + 1)}$

Step 6.7

Rewrite $- n {(n + 1)}^{\frac{1}{2}} + n^{\frac{3}{2}} + n^{\frac{1}{2}}$ in a factored form.

Tap for more steps...

Step 6.7.1

Factor $n^{\frac{1}{2}}$ out of $- n {(n + 1)}^{\frac{1}{2}} + n^{\frac{3}{2}} + n^{\frac{1}{2}}$ .

Tap for more steps...

Step 6.7.1.1

Factor $n^{\frac{1}{2}}$ out of $- n {(n + 1)}^{\frac{1}{2}}$ .

$\frac{n^{\frac{1}{2}} (- n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}}) + n^{\frac{3}{2}} + n^{\frac{1}{2}}}{n (n + 1)}$

Step 6.7.1.2

Factor $n^{\frac{1}{2}}$ out of $n^{\frac{3}{2}}$ .

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

Step 6.7.1.3

Multiply by $1$ .

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

Step 6.7.1.4

Factor $n^{\frac{1}{2}}$ out of $n^{\frac{1}{2}} (- n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}}) + n^{\frac{1}{2}} n^{\frac{2}{2}}$ .

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

Step 6.7.1.5

Factor $n^{\frac{1}{2}}$ out of $n^{\frac{1}{2}} (- n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}} + n^{\frac{2}{2}}) + n^{\frac{1}{2}} \cdot 1$ .

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

Step 6.7.2

Divide $2$ by $2$ .

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

Step 6.7.3

Simplify.

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

Step 7

Move $n^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 8

Multiply $n$ by $n^{- \frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 8.1

Move $n^{- \frac{1}{2}}$ .

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

Step 8.2

Multiply $n^{- \frac{1}{2}}$ by $n$ .

Tap for more steps...

Step 8.2.1

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

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

Step 8.2.2

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

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

Step 8.3

Write $1$ as a fraction with a common denominator.

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 8.5

Add $- 1$ and $2$ .

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

Step 9

Factor $- 1$ out of $- n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}}$ .

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

Step 10

Factor $- 1$ out of $n$ .

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

Step 11

Factor $- 1$ out of $- (n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}}) - 1 (- n)$ .

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

Step 12

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 13

Factor $- 1$ out of $- (n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}} - n) - 1 (- 1)$ .

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

Step 14

Rewrite $- (n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}} - n - 1)$ as $- 1 (n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}} - n - 1)$ .

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

Step 15

Move the negative in front of the fraction.

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

Step 16

Rewrite the summation.

$\sum_{n = 4}^{\infty} - \frac{n^{\frac{1}{2}} {(n + 1)}^{\frac{1}{2}} - n - 1}{n^{\frac{1}{2}} (n + 1)}$