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Question
\sum _{n=0}^{+\infty}\frac{n+2}{2n+7}
Determine the convergence or divergence
-
Determine the convergence or divergence using the nth Term Test
-
Determine the convergence or divergence using the Ratio Test
-
Determine the convergence or divergence using the Integral Test
\textrm{Diverges}
Evaluate
\sum _{n=0}^{+\infty}\frac{n+2}{2n+7}
Find the limit
\lim _{n\rightarrow +\infty}\left(\frac{n+2}{2n+7}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(\frac{n\left(1+\frac{2}{n}\right)}{n\left(2+\frac{7}{n}\right)}\right)
Reduce the fraction
\lim _{n\rightarrow +\infty}\left(\frac{1+\frac{2}{n}}{2+\frac{7}{n}}\right)
Rewrite the expression
\frac{\lim _{n\rightarrow +\infty}\left(1+\frac{2}{n}\right)}{\lim _{n\rightarrow +\infty}\left(2+\frac{7}{n}\right)}
Calculate
More Steps
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
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{2}{n}\right)
Rewrite the expression
\frac{2}{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
\frac{2}{+\infty}
Calculate
0
1+0
Calculate
1
\frac{1}{\lim _{n\rightarrow +\infty}\left(2+\frac{7}{n}\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(2+\frac{7}{n}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(2\right)+\lim _{n\rightarrow +\infty}\left(\frac{7}{n}\right)
Calculate
2+\lim _{n\rightarrow +\infty}\left(\frac{7}{n}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{7}{n}\right)
Rewrite the expression
\frac{7}{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
\frac{7}{+\infty}
Calculate
0
2+0
Calculate
2
\frac{1}{2}
Solution
\textrm{Diverges}
\textrm{Inconclusive}
Evaluate
\sum _{n=0}^{+\infty}\frac{n+2}{2n+7}
Find the limit
\lim _{n\rightarrow +\infty}\left(\left|\frac{\frac{n+3}{2n+9}}{\frac{n+2}{2n+7}}\right|\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\left|\frac{\left(n+3\right)\left(2n+7\right)}{\left(2n+9\right)\left(n+2\right)}\right|\right)
Remove the absolute value bars
\lim _{n\rightarrow +\infty}\left(\frac{\left(n+3\right)\left(2n+7\right)}{\left(2n+9\right)\left(n+2\right)}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(\frac{n^{2}\left(2+\frac{13}{n}+\frac{21}{n^{2}}\right)}{n^{2}\left(2+\frac{13}{n}+\frac{18}{n^{2}}\right)}\right)
Reduce the fraction
\lim _{n\rightarrow +\infty}\left(\frac{2+\frac{13}{n}+\frac{21}{n^{2}}}{2+\frac{13}{n}+\frac{18}{n^{2}}}\right)
Rewrite the expression
\frac{\lim _{n\rightarrow +\infty}\left(2+\frac{13}{n}+\frac{21}{n^{2}}\right)}{\lim _{n\rightarrow +\infty}\left(2+\frac{13}{n}+\frac{18}{n^{2}}\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(2+\frac{13}{n}+\frac{21}{n^{2}}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(2\right)+\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)+\lim _{n\rightarrow +\infty}\left(\frac{21}{n^{2}}\right)
Calculate
2+\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)+\lim _{n\rightarrow +\infty}\left(\frac{21}{n^{2}}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)
Rewrite the expression
\frac{13}{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
\frac{13}{+\infty}
Calculate
0
2+0+\lim _{n\rightarrow +\infty}\left(\frac{21}{n^{2}}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{21}{n^{2}}\right)
Rewrite the expression
\frac{21}{\lim _{n\rightarrow +\infty}\left(n^{2}\right)}
Calculate
\frac{21}{+\infty}
Calculate
0
2+0+0
Calculate
2+0
Calculate
2
\frac{2}{\lim _{n\rightarrow +\infty}\left(2+\frac{13}{n}+\frac{18}{n^{2}}\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(2+\frac{13}{n}+\frac{18}{n^{2}}\right)
Rewrite the expression
\lim _{n\rightarrow +\infty}\left(2\right)+\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)+\lim _{n\rightarrow +\infty}\left(\frac{18}{n^{2}}\right)
Calculate
2+\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)+\lim _{n\rightarrow +\infty}\left(\frac{18}{n^{2}}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{13}{n}\right)
Rewrite the expression
\frac{13}{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
\frac{13}{+\infty}
Calculate
0
2+0+\lim _{n\rightarrow +\infty}\left(\frac{18}{n^{2}}\right)
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(\frac{18}{n^{2}}\right)
Rewrite the expression
\frac{18}{\lim _{n\rightarrow +\infty}\left(n^{2}\right)}
Calculate
\frac{18}{+\infty}
Calculate
0
2+0+0
Calculate
2+0
Calculate
2
\frac{2}{2}
Calculate
1
Solution
\textrm{Inconclusive}
\text{Method not applicable}
Evaluate
\sum _{n=0}^{+\infty}\frac{n+2}{2n+7}
\text{Write the n-th term }\frac{n+2}{2n+7}\text{ as a function}
f\left(x\right)=\frac{x+2}{2x+7}
Rewrite the expression
\frac{d}{dx}\left(\frac{x+2}{2x+7}\right)
\text{Use differentiation rule }\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{\frac{d}{dx}(f(x))\times g(x)-f(x)\times \frac{d}{dx}(g\left(x)\right)}{\left(g(x)\right)^2}
\frac{\frac{d}{dx}\left(x+2\right)\times \left(2x+7\right)-\left(x+2\right)\times \frac{d}{dx}\left(2x+7\right)}{\left(2x+7\right)^{2}}
Calculate
More Steps
Evaluate
\frac{d}{dx}\left(x+2\right)
\text{Use differentiation rule }\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)\pm \frac{d}{dx}(g(x))
\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(2\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
1+\frac{d}{dx}\left(2\right)
\text{Use }\frac{d}{dx}(c)=0\text{ to find derivative}
1+0
Removing 0 doesn't change the value,so remove it from the expression
1
\frac{1\times \left(2x+7\right)-\left(x+2\right)\times \frac{d}{dx}\left(2x+7\right)}{\left(2x+7\right)^{2}}
Calculate
More Steps
Evaluate
\frac{d}{dx}\left(2x+7\right)
\text{Use differentiation rule }\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}\left(f\left(x\right)\right)\pm \frac{d}{dx}(g(x))
\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(7\right)
Calculate
More Steps
Calculate
\frac{d}{dx}\left(2x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
2\times \frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
2\times 1
Any expression multiplied by 1 remains the same
2
2+\frac{d}{dx}\left(7\right)
\text{Use }\frac{d}{dx}(c)=0\text{ to find derivative}
2+0
Removing 0 doesn't change the value,so remove it from the expression
2
\frac{1\times \left(2x+7\right)-\left(x+2\right)\times 2}{\left(2x+7\right)^{2}}
Any expression multiplied by 1 remains the same
\frac{2x+7-\left(x+2\right)\times 2}{\left(2x+7\right)^{2}}
Multiply the terms
\frac{2x+7-2\left(x+2\right)}{\left(2x+7\right)^{2}}
Calculate the sum or difference
More Steps
Evaluate
2x+7-2\left(x+2\right)
Expand the expression
More Steps
Calculate
-2\left(x+2\right)
Apply the distributive property
-2x-2\times 2
Multiply the numbers
-2x-4
2x+7-2x-4
The sum of two opposites equals 0
More Steps
Evaluate
2x-2x
Collect like terms
\left(2-2\right)x
Add the coefficients
0\times x
Calculate
0
0+7-4
Remove 0
7-4
Subtract the numbers
3
\frac{3}{\left(2x+7\right)^{2}}
Rewrite the expression
\frac{3}{\left(2x+7\right)^{2}}<0
\text{Since the left-hand side is always positive,and the right-hand side is always 0,the statement is false for any value of }x
x \notin \mathbb{R}
Solution
\text{Method not applicable}
Show Solutions
