1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Pavlova-9 [17]
3 years ago
12

Test for convergence

Mathematics
1 answer:
Elan Coil [88]3 years ago
4 0
You've established that

\displaystyle\sum_{n\ge1}\frac1{2^n}\le\int_1^\infty\frac{\mathrm dx}{2^x}

so all you need to do is compute the integral. Rewrite the integrand as

\dfrac1{2^x}=2^{-x}=e^{\ln2^{-x}}=e^{-x\ln2}

and replace y=-x\ln2, so that \mathrm dy=-\ln2\,\mathrm dx. Then

\displaystyle\int_{x=1}^{x\to\infty}\frac{\mathrm dx}{2^x}=\int_{y=-\ln2}^{y\to-\infty}e^y\,\frac{\mathrm dy}{-\ln 2}=\frac1{\ln2}\int_{-\infty}^{-\ln2}e^y\,\mathrm dy

By the fundamental theorem of calculus, this evaluates to

\dfrac1{\ln2}e^y\bigg|_{y\to-\infty}^{y=-\ln2}=\dfrac1{\ln2}\left(e^{-\ln2}-\lim_{y\to-\infty}e^y\right)=\dfrac{e^{-\ln2}}{\ln2}=\dfrac{\frac12}{\ln2}=-\dfrac1{2\ln2}=\dfrac1{\ln4}

and so the series converges.

You also could have used the fact that the series is geometric with common ratio less than 1 to arrive at the same conclusion, with the added perk of being able to find the exact value of the sum to corroborate this.
You might be interested in
If f(x)=g(x)+k, describe the relationship between f(x) and g(x).
Alecsey [184]
K is constant. if you increase g(x) by x. f(x) also increases by x
5 0
3 years ago
Use the table at the right.<br> a. What is the approximate volume of the small truck?
irina [24]

Volume of the small truck = 554.54 cubic ft

Solution:

Given data:

Length of the small truck = 11\frac{1}{13} ft

Width of the small truck = 7\frac{5}{12} ft

Height of the small truck = 6\frac{3}{4} ft

Volume of the small truck = length × width × height

                                           $=11\frac{1}{13}\times7\frac{5}{12}\times 6\frac{3}{4}

Let us change the mixed fraction into improper fraction.

                                          $=\frac{11\times 13 + 1}{13}\times\frac{7 \times 12 + 5}{12}\times \frac{6 \times 4 + 3}{4}

                                          $=\frac{144}{13}\times\frac{89}{12}\times \frac{27}{4}

                                          $=\frac{7209}{13}

                                          = 554. 54 cubic ft

Volume of the small truck = 554.54 cubic ft

6 0
3 years ago
Consider that an = 4n-1 represents a sequence. What is the tenth term?
Korvikt [17]
4×10-1
40-1
39 is your answer
7 0
3 years ago
How do i add <img src="https://tex.z-dn.net/?f=-%5Cfrac%7B3%7D%7B10%7D%20plus%20%5Cfrac%7B2%7D%7B3%20%7D%5C%5C" id="TexFormula1"
antoniya [11.8K]

-\dfrac{3}{10}+\dfrac{2}{3}=\dfrac{2}{3}+\left(-\dfrac{3}{10}\right)=\dfrac{2}{3}-\dfrac{3}{10}\\\\\text{We must find the common denominator.}\\\\Multiples\ of\ 3:0,\ 3,\ 6,\ 9,\ 12,\ 15,\ 18,\ 21,\ 24,\ 27,\ \boxed{30},\ 33,\ 36,...\\\\Multiples\ of\ 10:\ 0,\ 10,\ 20,\ \boxed{30},\ 40,\ 50,\ 60,...\\\\\dfrac{2}{3}=\dfrac{2\cdot10}{3\cdot10}=\dfrac{20}{30}\\\\\dfrac{3}{10}=\dfrac{3\cdot3}{10\cdot3}=\dfrac{9}{30}\\\\\dfrac{2}{3}-\dfrac{3}{10}=\dfrac{20}{30}-\dfrac{9}{30}=\dfrac{20-9}{30}=\dfrac{11}{30}

4 0
3 years ago
Help me and i will friend u
barxatty [35]

Answer:

ii.

Step-by-step explanation:

4 0
3 years ago
Read 2 more answers
Other questions:
  • Help me. I'm not very good at math
    8·1 answer
  • Choose the correct simplification of the expression (6x2y)2(y2)3.
    5·2 answers
  • Which statement best describes the graph of x3 − 3x2 − x + 3?
    13·2 answers
  • Yesterday 96 people attended the school band concert, and 3/4 of them stayed for the cookies and punch. How many people stayed f
    6·1 answer
  • Yo got an easy questioin for u guys why does canvas suck so much?
    14·2 answers
  • Charlie and Alice drove to Miami. The total distance was 120 miles. Charlie drove 90 of the miles.
    8·2 answers
  • Solve the equation below:<br> 1/2y - 9 &gt; 1
    7·1 answer
  • Help I need it plssssssssss
    8·2 answers
  • Find the sum. Write your answer in standard form.<br> (7x +13) + (x3 + 4x2 – 8)
    7·2 answers
  • Which shape listen below will result from cross section b?
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!