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Question
\sum _{n=0}^{+\infty}\frac{3}{2^{n}}
Determine the convergence or divergence
-
Determine the convergence or divergence using the nth Term Test
-
Determine the convergence or divergence using the Geometric Series Test
-
Determine the convergence or divergence using the Ratio Test
-
Determine the convergence or divergence using the Integral Test
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\textrm{Inconclusive}
Evaluate
\sum _{n=0}^{+\infty}\frac{3}{2^{n}}
Find the limit
\lim _{n\rightarrow +\infty}\left(\frac{3}{2^{n}}\right)
Rewrite the expression
\frac{3}{\lim _{n\rightarrow +\infty}\left(2^{n}\right)}
Calculate
More Steps
Evaluate
\lim _{n\rightarrow +\infty}\left(2^{n}\right)
Rewrite the expression
2^{\lim _{n\rightarrow +\infty}\left(n\right)}
Calculate
2^{+\infty}
Calculate
+\infty
\frac{3}{+\infty}
Calculate
0
Solution
\textrm{Inconclusive}
\text{Converges}
Evaluate
\sum _{n=0}^{+\infty}\frac{3}{2^{n}}
Find the common ratio
\frac{1}{2}
Solution
\text{Converges}
\text{Converges}
Evaluate
\sum _{n=0}^{+\infty}\frac{3}{2^{n}}
Find the limit
\lim _{n\rightarrow +\infty}\left(\left|\frac{\frac{3}{2^{n+1}}}{\frac{3}{2^{n}}}\right|\right)
Simplify
\lim _{n\rightarrow +\infty}\left(\left|\frac{1}{2}\right|\right)
Remove the absolute value bars
\lim _{n\rightarrow +\infty}\left(\frac{1}{2}\right)
Calculate
\frac{1}{2}
Solution
\text{Converges}
\text{Converges}
Evaluate
\sum _{n=0}^{+\infty}\frac{3}{2^{n}}
\text{Write the n-th term }\frac{3}{2^{n}}\text{ as a function}
f\left(x\right)=\frac{3}{2^{x}}
Rewrite the expression
\frac{d}{dx}\left(\frac{3}{2^{x}}\right)
Use differentiation rules
3\times \frac{d}{dx}\left(\frac{1}{2^{x}}\right)
Rewrite the expression in exponential form
3\times \frac{d}{dx}\left(2^{-x}\right)
Calculate the derivative
More Steps
Evaluate
\frac{d}{dx}\left(2^{-x}\right)
\text{Use the chain rule }\frac{d}{dx}(f(g))=\frac{d}{dg}(f(g))\times \frac{d}{dx}(g)\text{ where the }g\text{=}-x\text{, to find the derivative}
\frac{d}{dg}\left(2^{g}\right)\times \frac{d}{dx}\left(-x\right)
\text{Use }\frac{d}{dx} a^{x}=\ln(a)a^{x}\text{ to find derivative}
\ln{\left(2\right)}\times 2^{g}\times \frac{d}{dx}\left(-x\right)
Calculate
More Steps
Calculate
\frac{d}{dx}\left(-x\right)
\text{Use differentiation rule }\frac{d}{dx}\left(cf\left(x\right)\right)=c\times\frac{d}{dx}(f(x))
-\frac{d}{dx}\left(x\right)
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
-1
\ln{\left(2\right)}\times 2^{g}\left(-1\right)
Substitute back
\ln{\left(2\right)}\times 2^{-x}\left(-1\right)
Rewrite the expression
-\ln{\left(2\right)}\times 2^{-x}
3\left(-\ln{\left(2\right)}\times 2^{-x}\right)
Rewrite the expression
More Steps
Evaluate
-\ln{\left(2\right)}\times 2^{-x}
\text{Express with a positive exponent using }a^{-n}=\frac{1}{a^n}
-\ln{\left(2\right)}\times \frac{1}{2^{x}}
Rewrite the expression
\frac{-\ln{\left(2\right)}}{2^{x}}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{\ln{\left(2\right)}}{2^{x}}
3\left(-\frac{\ln{\left(2\right)}}{2^{x}}\right)
Multiply the terms
-\frac{3\ln{\left(2\right)}}{2^{x}}
Rewrite the expression
-\frac{3\ln{\left(2\right)}}{2^{x}}<0
\text{Since the left-hand side is always negative,and the right-hand side is always 0,the statement is true for any value of }x
x \in \mathbb{R}
\text{Since the function }f(x)=\frac{3}{2^{x}}\text{ is positive,continuous,and decreasing for }x \ge 0\text{,use the Integral Test}
\int _{0}^{+\infty}\frac{3}{2^{x}} dx
By definition,rewrite the improper integral using one-sided limit and a definite integral
\lim _{a\rightarrow +\infty}\left(\int _{0}^{a}\frac{3}{2^{x}} dx\right)
Evaluate the integral
More Steps
Evaluate
\int _{0}^{a}\frac{3}{2^{x}} dx
Evaluate the integral
\int \frac{3}{2^{x}} dx
Rewrite the expression
\int 3\times 2^{-x} dx
\text{Use the property of integral }\int kf(x) dx=k\int f(x) dx
3\times \int 2^{-x} dx
\text{Use the substitution }dx=-1 dt\text{ to transform the integral}
More Steps
Evaluate
t=-x
Calculate the derivative
dt=-1dx
Evaluate
dx=-1dt
3\times \int 2^{-x}\left(-1\right) dt
Simplify
3\times \int -2^{-x} dt
\text{Use the substitution }t=-x\text{ to transform the integral}
3\times \int -2^{t} dt
\text{Use the property of integral }\int kf(x) dx=k\int f(x) dx
3\left(-1\right)\times \int 2^{t} dt
Simplify
-3\times \int 2^{t} dt
\text{Use the property of integral }\int a^{x} dx = \frac{ a^{x}}{\ln a}
-3\times \frac{2^{t}}{\ln{\left(2\right)}}
Multiply the terms
-\frac{3\times 2^{t}}{\ln{\left(2\right)}}
Substitute back
-\frac{3\times 2^{-x}}{\ln{\left(2\right)}}
Divide the terms
More Steps
Evaluate
\frac{3\times 2^{-x}}{\ln{\left(2\right)}}
Rewrite the expression
\frac{\frac{3}{2^{x}}}{\ln{\left(2\right)}}
Multiply by the reciprocal
\frac{3}{2^{x}}\times \frac{1}{\ln{\left(2\right)}}
Multiply the terms
\frac{3}{2^{x}\ln{\left(2\right)}}
Multiply the terms
\frac{3}{\ln{\left(2\right)}\times 2^{x}}
-\frac{3}{\ln{\left(2\right)}\times 2^{x}}
Return the limits
\left(-\frac{3}{\ln{\left(2\right)}\times 2^{x}}\right)\bigg |_{0}^{a}
Substitute the values into formula
-\frac{3}{\ln{\left(2\right)}\times 2^{a}}-\left(-\frac{3}{\ln{\left(2\right)}\times 2^{0}}\right)
Evaluate the power
-\frac{3}{\ln{\left(2\right)}\times 2^{a}}-\left(-\frac{3}{\ln{\left(2\right)}\times 1}\right)
Any expression multiplied by 1 remains the same
-\frac{3}{\ln{\left(2\right)}\times 2^{a}}-\left(-\frac{3}{\ln{\left(2\right)}}\right)
Subtract the terms
-\frac{3}{\ln{\left(2\right)}\times 2^{a}}+\frac{3}{\ln{\left(2\right)}}
\lim _{a\rightarrow +\infty}\left(-\frac{3}{\ln{\left(2\right)}\times 2^{a}}+\frac{3}{\ln{\left(2\right)}}\right)
Rewrite the expression
\lim _{a\rightarrow +\infty}\left(-\frac{3}{\ln{\left(2\right)}\times 2^{a}}\right)+\lim _{a\rightarrow +\infty}\left(\frac{3}{\ln{\left(2\right)}}\right)
Calculate
More Steps
Evaluate
\lim _{a\rightarrow +\infty}\left(-\frac{3}{\ln{\left(2\right)}\times 2^{a}}\right)
Rewrite the expression
-\lim _{a\rightarrow +\infty}\left(\frac{3}{\ln{\left(2\right)}\times 2^{a}}\right)
Calculate
More Steps
Evaluate
\lim _{a\rightarrow +\infty}\left(\frac{3}{\ln{\left(2\right)}\times 2^{a}}\right)
Rewrite the expression
\frac{3}{\lim _{a\rightarrow +\infty}\left(\ln{\left(2\right)}\times 2^{a}\right)}
Calculate
\frac{3}{+\infty}
Calculate
0
0
0+\lim _{a\rightarrow +\infty}\left(\frac{3}{\ln{\left(2\right)}}\right)
Calculate
0+\frac{3}{\ln{\left(2\right)}}
Calculate
\frac{3}{\ln{\left(2\right)}}
Solution
\text{Converges}
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