Determine Convergence with the Integral Test

$\sum_{n = 1}^{\infty} \frac{1}{1 + n^{2}}$

Step 1

To determine if the series is convergent, determine if the integral of the sequence is convergent.

$\int_{1}^{\infty} \frac{1}{1 + x^{2}} d x$

Step 2

Write the integral as a limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \int_{1}^{t} \frac{1}{1 + x^{2}} d x$

Step 3

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

$lim_{t \to \infty} \int_{1}^{t} \frac{1}{1^{2} + x^{2}} d x$

Step 4

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x)]_{1}^{t}$ .

$lim_{t \to \infty} \arctan (x)]_{1}^{t}$

Step 5

Simplify the answer.

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Step 5.1

Evaluate $\arctan (x)$ at $t$ and at $1$ .

$lim_{t \to \infty} (\arctan (t)) - \arctan (1)$

Step 5.2

Remove parentheses.

$lim_{t \to \infty} \arctan (t) - \arctan (1)$

Step 6

Evaluate the limit.

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Step 6.1

Split the limit using the Sum of Limits Rule on the limit as $t$ approaches $\infty$ .

$lim_{t \to \infty} \arctan (t) - lim_{t \to \infty} \arctan (1)$

Step 6.2

The limit as $t$ approaches $\infty$ is $\frac{π}{2}$ .

$\frac{π}{2} - lim_{t \to \infty} \arctan (1)$

Step 6.3

Evaluate the limit of $\arctan (1)$ which is constant as $t$ approaches $\infty$ .

$\frac{π}{2} - \arctan (1)$

Step 6.4

Simplify the answer.

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Step 6.4.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$\frac{π}{2} - \frac{π}{4}$

Step 6.4.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π}{2} \cdot \frac{2}{2} - \frac{π}{4}$

Step 6.4.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.3.1

Multiply $\frac{π}{2}$ by $\frac{2}{2}$ .

$\frac{π \cdot 2}{2 \cdot 2} - \frac{π}{4}$

Step 6.4.3.2

Multiply $2$ by $2$ .

$\frac{π \cdot 2}{4} - \frac{π}{4}$

Step 6.4.4

Combine the numerators over the common denominator.

$\frac{π \cdot 2 - π}{4}$

Step 6.4.5

Simplify the numerator.

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Step 6.4.5.1

Move $2$ to the left of $π$ .

$\frac{2 \cdot π - π}{4}$

Step 6.4.5.2

Subtract $π$ from $2 π$ .

$\frac{π}{4}$

Step 7

Since the integral is convergent, the series is convergent.

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Determine Convergence with the Integral Test

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