Calculus Examples

\infty \sum n = 1 {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n}

$\sum_{n = 1}^{\infty} {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n}$

Step 1

For an infinite series

\sum a_{n}

$\sum_{}^{} a_{n}$ , find the limit

L = lim n \to \infty {| a_{n} |}^{\frac{1}{n}}

$L = lim_{n \to \infty} {| a_{n} |}^{\frac{1}{n}}$ to determine convergence using Cauchy's Root Test.

L = lim n \to \infty {| a_{n} |}^{\frac{1}{n}}

$L = lim_{n \to \infty} {| a_{n} |}^{\frac{1}{n}}$

Step 2

Substitute for

a_{n}

$a_{n}$ .

L = lim n \to \infty {∣ ∣ ∣ {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n} ∣ ∣ ∣}^{\frac{1}{n}}

$L = lim_{n \to \infty} {| {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n} |}^{\frac{1}{n}}$

Step 3

Simplify.

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Step 3.1

Move the exponent into the absolute value.

L = lim n \to \infty ∣ ∣ ∣ ∣ {({(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n})}^{\frac{1}{n}} ∣ ∣ ∣ ∣

$L = lim_{n \to \infty} | {({(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n})}^{\frac{1}{n}} |$

Step 3.2

Multiply the exponents in

{({(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n})}^{\frac{1}{n}}

${({(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n})}^{\frac{1}{n}}$ .

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Step 3.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

L = lim n \to \infty ∣ ∣ ∣ ∣ {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n \frac{1}{n}} ∣ ∣ ∣ ∣

$L = lim_{n \to \infty} | {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n \frac{1}{n}} |$

Step 3.2.2

Cancel the common factor of

$n$ .

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Step 3.2.2.1

Cancel the common factor.

$L = lim_{n \to \infty} | {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{n \frac{1}{n}} |$

Step 3.2.2.2

Rewrite the expression.

$L = lim_{n \to \infty} | {(\frac{2 n + n^{3}}{5 n^{3} + 1})}^{1} |$

Step 3.3

Simplify.

$L = lim_{n \to \infty} | \frac{2 n + n^{3}}{5 n^{3} + 1} |$

Step 4

Evaluate the limit.

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Step 4.1

Move the limit inside the absolute value signs.

$L = | lim_{n \to \infty} \frac{2 n + n^{3}}{5 n^{3} + 1} |$

Step 4.2

Divide the numerator and denominator by the highest power of

$n$ in the denominator, which is

$n^{3}$ .

$L = | lim_{n \to \infty} \frac{\frac{2 n}{n^{3}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3

Evaluate the limit.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Cancel the common factor of

$n$ and

$n^{3}$ .

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Step 4.3.1.1.1

Factor

$n$ out of

$2 n$ .

$L = | lim_{n \to \infty} \frac{\frac{n \cdot 2}{n^{3}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.1.1.2

Cancel the common factors.

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Step 4.3.1.1.2.1

Factor

$n$ out of

$n^{3}$ .

$L = | lim_{n \to \infty} \frac{\frac{n \cdot 2}{n \cdot n^{2}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.1.1.2.2

Cancel the common factor.

$L = | lim_{n \to \infty} \frac{\frac{n \cdot 2}{n \cdot n^{2}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.1.1.2.3

Rewrite the expression.

$L = | lim_{n \to \infty} \frac{\frac{2}{n^{2}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.1.2

Cancel the common factor of

$n^{3}$ .

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Step 4.3.1.2.1

Cancel the common factor.

$L = | lim_{n \to \infty} \frac{\frac{2}{n^{2}} + \frac{n^{3}}{n^{3}}}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.1.2.2

Rewrite the expression.

$L = | lim_{n \to \infty} \frac{\frac{2}{n^{2}} + 1}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.2

Cancel the common factor of

$n^{3}$ .

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Step 4.3.2.1

Cancel the common factor.

$L = | lim_{n \to \infty} \frac{\frac{2}{n^{2}} + 1}{\frac{5 n^{3}}{n^{3}} + \frac{1}{n^{3}}} |$

Step 4.3.2.2

Divide

$5$ by

$1$ .

$L = | lim_{n \to \infty} \frac{\frac{2}{n^{2}} + 1}{5 + \frac{1}{n^{3}}} |$

Step 4.3.3

Split the limit using the Limits Quotient Rule on the limit as

$n$ approaches

$\infty$ .

$L = | \frac{lim_{n \to \infty} \frac{2}{n^{2}} + 1}{lim_{n \to \infty} 5 + \frac{1}{n^{3}}} |$

Step 4.3.4

Split the limit using the Sum of Limits Rule on the limit as

$n$ approaches

$\infty$ .

$L = | \frac{lim_{n \to \infty} \frac{2}{n^{2}} + lim_{n \to \infty} 1}{lim_{n \to \infty} 5 + \frac{1}{n^{3}}} |$

Step 4.3.5

Move the term

$2$ outside of the limit because it is constant with respect to

$n$ .

$L = | \frac{2 lim_{n \to \infty} \frac{1}{n^{2}} + lim_{n \to \infty} 1}{lim_{n \to \infty} 5 + \frac{1}{n^{3}}} |$

Step 4.4

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{n^{2}}$ approaches

$0$ .

$L = | \frac{2 \cdot 0 + lim_{n \to \infty} 1}{lim_{n \to \infty} 5 + \frac{1}{n^{3}}} |$

Step 4.5

Evaluate the limit.

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Step 4.5.1

Evaluate the limit of

$1$ which is constant as

$n$ approaches

$\infty$ .

$L = | \frac{2 \cdot 0 + 1}{lim_{n \to \infty} 5 + \frac{1}{n^{3}}} |$

Step 4.5.2

Split the limit using the Sum of Limits Rule on the limit as

$n$ approaches

$\infty$ .

$L = | \frac{2 \cdot 0 + 1}{lim_{n \to \infty} 5 + lim_{n \to \infty} \frac{1}{n^{3}}} |$

Step 4.5.3

Evaluate the limit of

$5$ which is constant as

$n$ approaches

$\infty$ .

$L = | \frac{2 \cdot 0 + 1}{5 + lim_{n \to \infty} \frac{1}{n^{3}}} |$

Step 4.6

Since its numerator approaches a real number while its denominator is unbounded, the fraction

$\frac{1}{n^{3}}$ approaches

$0$ .

$L = | \frac{2 \cdot 0 + 1}{5 + 0} |$

Step 4.7

Simplify the answer.

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Multiply

$2$ by

$0$ .

$L = | \frac{0 + 1}{5 + 0} |$

Step 4.7.1.2

Add

$0$ and

$1$ .

$L = | \frac{1}{5 + 0} |$

Step 4.7.2

Add

$5$ and

$0$ .

$L = | \frac{1}{5} |$

Step 4.7.3

$\frac{1}{5}$ is approximately

$0.2$ which is positive so remove the absolute value

$L = \frac{1}{5}$

Step 4.8

Divide

$1$ by

$5$ .

$L = 0.2$

Step 5

$L < 1$ , the series is absolutely convergent. If

$L > 1$ , the series is divergent. If

$L = 1$ , the test is inconclusive. In this case,

$L < 1$ .

The series is convergent on

$[1, \infty)$

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