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Question
\sum _{n=0}^{+\infty}\frac{\left(-1\right)^{n}}{n^{2}+1}
Determine the convergence or divergence
\text{Converges}
Evaluate
\sum _{n=0}^{+\infty}\frac{\left(-1\right)^{n}}{n^{2}+1}
Find the limit
\lim _{n\rightarrow +\infty}\left(\left|\frac{\left(-1\right)^{n}}{n^{2}+1}\right|\right)
Remove the absolute value bars
\lim _{n\rightarrow +\infty}\left(\frac{1}{n^{2}+1}\right)
Rewrite the expression
\frac{1}{\lim _{n\rightarrow +\infty}\left(n^{2}+1\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(n^{2}+1\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(n^{2}\right)+\lim _{n\rightarrow +\infty}\left(1\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(n^{2}\right)
Rewrite the expression
\left(\lim _{n\rightarrow +\infty}\left(n\right)\right)^{2}
Calculate
\left(+\infty\right)^{2}
Calculate
+\infty
\left(+\infty\right)+\lim _{n\rightarrow +\infty}\left(1\right)
Calculate
\left(+\infty\right)+1
Calculate
+\infty
\frac{1}{+\infty}
Calculate
0
Rewrite the expression
-\frac{n^{2}+1}{n^{2}+2n+2}>-1
Change the signs on both sides of the inequality and flip the inequality sign
\frac{n^{2}+1}{n^{2}+2n+2}<1
Calculate
\frac{n^{2}+1}{n^{2}+2n+2}-1<0
Calculate
More Steps
Calculate
\frac{n^{2}+1}{n^{2}+2n+2}-1
Reduce fractions to a common denominator
\frac{n^{2}+1}{n^{2}+2n+2}-\frac{n^{2}+2n+2}{n^{2}+2n+2}
Write all numerators above the common denominator
\frac{n^{2}+1-\left(n^{2}+2n+2\right)}{n^{2}+2n+2}
Calculate the sum or difference
More Steps
Evaluate
n^{2}+1-\left(n^{2}+2n+2\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
n^{2}+1-n^{2}-2n-2
The sum of two opposites equals 0
0+1-2n-2
Remove 0
1-2n-2
Subtract the numbers
-1-2n
\frac{-1-2n}{n^{2}+2n+2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{1+2n}{n^{2}+2n+2}
-\frac{1+2n}{n^{2}+2n+2}<0
Change the signs on both sides of the inequality and flip the inequality sign
\frac{1+2n}{n^{2}+2n+2}>0
\text{Separate the inequality into }2\text{ possible cases}
\begin{align}&\left\{ \begin{array}{l}1+2n>0\\n^{2}+2n+2>0\end{array}\right.\\&\left\{ \begin{array}{l}1+2n<0\\n^{2}+2n+2<0\end{array}\right.\end{align}
Solve the inequality
More Steps
Evaluate
1+2n>0
Move the constant to the right side
2n>0-1
Removing 0 doesn't change the value,so remove it from the expression
2n>-1
Divide both sides
\frac{2n}{2}>\frac{-1}{2}
Divide the numbers
n>\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
n>-\frac{1}{2}
\begin{align}&\left\{ \begin{array}{l}n>-\frac{1}{2}\\n^{2}+2n+2>0\end{array}\right.\\&\left\{ \begin{array}{l}1+2n<0\\n^{2}+2n+2<0\end{array}\right.\end{align}
Solve the inequality
More Steps
Evaluate
n^{2}+2n+2>0
Move the constant to the right side
n^{2}+2n>0-2
Add the terms
n^{2}+2n>-2
Add the same value to both sides
n^{2}+2n+1>-2+1
Evaluate
n^{2}+2n+1>-1
Evaluate
\left(n+1\right)^{2}>-1
Calculate
n \in \mathbb{R}
\begin{align}&\left\{ \begin{array}{l}n>-\frac{1}{2}\\n \in \mathbb{R}\end{array}\right.\\&\left\{ \begin{array}{l}1+2n<0\\n^{2}+2n+2<0\end{array}\right.\end{align}
Solve the inequality
More Steps
Evaluate
1+2n<0
Move the constant to the right side
2n<0-1
Removing 0 doesn't change the value,so remove it from the expression
2n<-1
Divide both sides
\frac{2n}{2}<\frac{-1}{2}
Divide the numbers
n<\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
n<-\frac{1}{2}
\begin{align}&\left\{ \begin{array}{l}n>-\frac{1}{2}\\n \in \mathbb{R}\end{array}\right.\\&\left\{ \begin{array}{l}n<-\frac{1}{2}\\n^{2}+2n+2<0\end{array}\right.\end{align}
Solve the inequality
More Steps
Evaluate
n^{2}+2n+2<0
Move the constant to the right side
n^{2}+2n<0-2
Add the terms
n^{2}+2n<-2
Add the same value to both sides
n^{2}+2n+1<-2+1
Evaluate
n^{2}+2n+1<-1
Evaluate
\left(n+1\right)^{2}<-1
Calculate
n \notin \mathbb{R}
\begin{align}&\left\{ \begin{array}{l}n>-\frac{1}{2}\\n \in \mathbb{R}\end{array}\right.\\&\left\{ \begin{array}{l}n<-\frac{1}{2}\\n \notin \mathbb{R}\end{array}\right.\end{align}
Find the intersection
\begin{align}&n>-\frac{1}{2}\\&\left\{ \begin{array}{l}n<-\frac{1}{2}\\n \notin \mathbb{R}\end{array}\right.\end{align}
Find the intersection
\begin{align}&n>-\frac{1}{2}\\&n \notin \mathbb{R}\end{align}
Find the union
n>-\frac{1}{2}
Calculate
\frac{1}{n^{2}+1}
Calculate
\frac{1}{n^{2}+2n+2}
The inequality is true
\frac{1}{n^{2}+1}>\frac{1}{n^{2}+2n+2}
Solution
\text{Converges}
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