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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.
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Step-by-step explanation:

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2 years ago
What is the area of this parallelogram?
kirza4 [7]

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Read 2 more answers
Isosceles JAM has convex angle M. JM=35,MA=4n+3, angle J=14w-1, and angle m=98 Find n. Find w. Find the perimeter of JAM
Lena [83]

Answer:

n = 8, w = 3 and perimeter = 122.83 units.

Step-by-step explanation:

Let the angle M is the angle between the equal sides of isosceles JAM.

So, JM = MA

⇒ 35 = 4n + 3

⇒ 4n = 32

⇒ n = 8 (Answer)

Now, if ∠ J = 14w - 1 and ∠ M = 98°, then

2(14w - 1) + 98 = 180

⇒ 2(14w - 1) = 82

⇒ 14w - 1 = 41

⇒ w = 3 (Answer)

Now, draw a perpendicular bisector on JA from vertex M and it meets JA at P say.

So, Δ MPJ will be a right triangle with ∠ J = (14w - 1) = 41° {Since w = 3}

Hence, \cos 41 = \frac{JP}{JM} = \frac{JP}{35}

⇒ JP = 35 cos 41 = 26.415

So, JA = 2 × JP = 52.83  

So, the perimeter of Δ JAM is = 35 × 2 + 52.83 = 122.83 units (Answer)

4 0
3 years ago
If (5⁴)m = 5 ¹²,what is the value of m<br> 2<br><br> 3<br><br> 5<br><br> 8
Lapatulllka [165]
The correct answer is:  [B]:  " 3 " .
_______________________________________________
Note:  
_______________________________________________
The question asks:
_______________________________________________
If (5⁴)ᵐ  = 5¹² ;  then what is "m" ?

Note the following property of exponents:
________________________________________________
   (xᵃ)ᵇ = x⁽ᵃ*ᵇ⁾ ; a≠0 ; b ≠ 0 ;  

Then,  (5⁴)³ = 5⁽⁴ * ³⁾ = 5¹² .

So, "12 ÷ 4 = 3" .
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The answer is:  "m = 3" ; which is:  Answer choice:  [B] .
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5 0
3 years ago
How to complete the problem
Fittoniya [83]
For step 2 it should be x^2-16x+73
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