\sum _{n=1}^{\infty}\frac{1}{n(n+1)} =
Question
\sum _{n=1}^{+\infty}\frac{1}{n\times \left(n+1\right)}
Determine the convergence or divergence
Determine the convergence or divergence using the nth Term Test
\text{Inconclusive}
Evaluate
\sum _{n=1}^{+\infty}\frac{1}{n\times \left(n+1\right)}
Multiply the terms
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Evaluate
n\times \left(n+1\right)
Apply the distributive property
n\times n+n\times 1
Multiply the terms
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Evaluate
n\times n
n^{1+1}
n^{2}
n^{2}+n\times 1
Multiply the terms
n^{2}+n
\sum _{n=1}^{+\infty}\frac{1}{n^{2}+n}
Find the limit
\lim _{n\rightarrow +\infty}\left(\frac{1}{n^{2}+n}\right)
Rewrite the expression
\frac{1}{\lim _{n\rightarrow +\infty}\left(n^{2}+n\right)}
Calculate
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Evaluate
\lim _{n\rightarrow +\infty}\left(n^{2}+n\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(n^{2}\right)+\lim _{n\rightarrow +\infty}\left(n\right)
Calculate
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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
+\infty+\lim _{n\rightarrow +\infty}\left(n\right)
Calculate
+\infty++\infty
Calculate
+\infty
\frac{1}{+\infty}
Calculate
0
Solution
\text{Inconclusive}
Determine the convergence or divergence using the Limit Comparison Test
\text{Converges}
Evaluate
\sum _{n=1}^{+\infty}\frac{1}{n\times \left(n+1\right)}
Multiply the terms
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Evaluate
n\times \left(n+1\right)
Apply the distributive property
n\times n+n\times 1
Multiply the terms
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Evaluate
n\times n
n^{1+1}
n^{2}
n^{2}+n\times 1
Multiply the terms
n^{2}+n
\sum _{n=1}^{+\infty}\frac{1}{n^{2}+n}
Calculate
\lim _{n\rightarrow +\infty}\left(\frac{\frac{1}{n^{2}+n}}{\frac{1}{n^{2}}}\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\frac{n}{n+1}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(\frac{n}{n}\times \frac{1}{1+\frac{1}{n}}\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\frac{1}{1+\frac{1}{n}}\right)
Rewrite the expression
\frac{1}{\lim _{n\rightarrow +\infty}\left(1+\frac{1}{n}\right)}
Rewrite the expression
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Evaluate
\lim _{n\rightarrow +\infty}\left(1+\frac{1}{n}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(1\right)+\lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Calculate
1+\lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Calculate
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Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Rewrite the expression
\frac{1}{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
\frac{1}{+\infty}
Calculate
0
1+0
Calculate
1
\frac{1}{1}
Divide the terms
1
Solution
\text{Converges}
Determine the convergence or divergence using the Ratio Test
\text{Inconclusive}
Evaluate
\sum _{n=1}^{+\infty}\frac{1}{n\times \left(n+1\right)}
Multiply the terms
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Evaluate
n\times \left(n+1\right)
Apply the distributive property
n\times n+n\times 1
Multiply the terms
More Steps Hide Steps
Evaluate
n\times n
n^{1+1}
n^{2}
n^{2}+n\times 1
Multiply the terms
n^{2}+n
\sum _{n=1}^{+\infty}\frac{1}{n^{2}+n}
Find the limit
\lim _{n\rightarrow +\infty}\left(\left|\frac{\frac{1}{n^{2}+3n+2}}{\frac{1}{n^{2}+n}}\right|\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\left|\frac{n}{n+2}\right|\right)
Remove the absolute value bars
\lim _{n\rightarrow +\infty}\left(\frac{n}{n+2}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(\frac{n}{n}\times \frac{1}{1+\frac{2}{n}}\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\frac{1}{1+\frac{2}{n}}\right)
Rewrite the expression
\frac{1}{\lim _{n\rightarrow +\infty}\left(1+\frac{2}{n}\right)}
Rewrite the expression
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Evaluate
\lim _{n\rightarrow +\infty}\left(1+\frac{2}{n}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(1\right)+\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Calculate
1+\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Calculate
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Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Rewrite the expression
2\times \lim _{n\rightarrow +\infty}\left(\frac{1}{n}\right)
Calculate
2\times 0
Calculate
0
1+0
Calculate
1
\frac{1}{1}
Divide the terms
1
Solution
\text{Inconclusive}
Determine the convergence or divergence using the Integral Test
\text{Converges}
Evaluate
\sum _{n=1}^{+\infty}\frac{1}{n\times \left(n+1\right)}
Multiply the terms
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Evaluate
n\times \left(n+1\right)
Apply the distributive property
n\times n+n\times 1
Multiply the terms
More Steps Hide Steps
Evaluate
n\times n
n^{1+1}
n^{2}
n^{2}+n\times 1
Multiply the terms
n^{2}+n
\sum _{n=1}^{+\infty}\frac{1}{n^{2}+n}
\text{Write the n-th term }\frac{1}{n^{2}+n}\text{ as a function}
f\left(x\right)=\frac{1}{x^{2}+x}
\text{Since the function f(}x\text{)=}\frac{1}{x^{2}+x}\text{ is positive,continuous,and decreasing for }x\text{>=}1\text{,use the Integral Test}
\int_{1}^{+\infty} \frac{1}{x^{2}+x} dx
Rewrite the improper integral
\lim _{a\rightarrow +\infty}\left(\int_{1}^{a} \frac{1}{x^{2}+x} dx\right)
Evaluate the integral
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Evaluate
\int_{1}^{a} \frac{1}{x^{2}+x} dx
Evaluate the integral
\int \frac{1}{x^{2}+x} dx
Evaluate the integral
\ln{\left(\left|x\right|\right)}-\ln{\left(\left|x+1\right|\right)}
Return the limits
\left(\ln{\left(\left|x\right|\right)}-\ln{\left(\left|x+1\right|\right)}\right)\bigg |_{1}^{a}
Substitute the values into formula
\ln{\left(\left|a\right|\right)}-\ln{\left(\left|a+1\right|\right)}-\left(\ln{\left(\left|1\right|\right)}-\ln{\left(\left|1+1\right|\right)}\right)
Evaluate
\ln{\left(\left|a\right|\right)}-\ln{\left(\left|a+1\right|\right)}-\left(-\ln{\left(2\right)}\right)
Calculate the value
\ln{\left(\left|a\right|\right)}-\ln{\left(\left|a+1\right|\right)}+\ln{\left(2\right)}
\lim _{a\rightarrow +\infty}\left(\ln{\left(\left|a\right|\right)}-\ln{\left(\left|a+1\right|\right)}+\ln{\left(2\right)}\right)
Rewrite the expression
\lim _{a\rightarrow +\infty}\left(\ln{\left(a\right)}-\ln{\left(a+1\right)}+\ln{\left(2\right)}\right)
Rearrange the terms
\lim _{a\rightarrow +\infty}\left(\ln{\left(\frac{2a}{a+1}\right)}\right)
Rewrite the expression
\ln{\left(\lim _{a\rightarrow +\infty}\left(\frac{2a}{a+1}\right)\right)}
Calculate
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Evaluate
\lim _{a\rightarrow +\infty}\left(\frac{2a}{a+1}\right)
Rewrite the expression
\lim _{a\rightarrow +\infty}\left(\frac{a}{a}\times \frac{2}{1+\frac{1}{a}}\right)
Simplify
\lim _{a\rightarrow +\infty}\left(\frac{2}{1+\frac{1}{a}}\right)
Rewrite the expression
\frac{2}{\lim _{a\rightarrow +\infty}\left(1+\frac{1}{a}\right)}
Rewrite the expression
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Evaluate
\lim _{a\rightarrow +\infty}\left(1+\frac{1}{a}\right)
Rewrite the expression
\lim _{a\rightarrow +\infty}\left(1\right)+\lim _{a\rightarrow +\infty}\left(\frac{1}{a}\right)
Calculate
1+\lim _{a\rightarrow +\infty}\left(\frac{1}{a}\right)
Calculate
1+0
Calculate
1
\frac{2}{1}
Divide the terms
2
\ln{\left(2\right)}
Solution
\text{Converges}
Choose Method
Determine the convergence or divergence using the nth Term Test
Determine the convergence or divergence using the Limit Comparison Test
Determine the convergence or divergence using the Ratio Test
Determine the convergence or divergence using the Integral Test