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A.If a new series is found by differentiating a known series with included endpoints, those endpoints might not be included in t

Aleonysh [2.5K]

Consider the power series

$\displaystyle\sum_{n\ge1}\dfrac{x^n}n$

By the ratio test, this series converges for

$\displaystyle\lim_{n\to\infty}\left|\frac{x^{n+1}}{n+1}\cdot\frac n{x^n}\right|=|x|\lim_{n\to\infty}\frac n{n+1}=|x|$

though we know by the alternating series test that the series converges for $x=-1$ .

So this series converges for $-1\le x$ .

Differentiating the series yields

$\displaystyle\frac{\mathrm d}{\mathrm dx}\sum_{n\ge1}\frac{x^n}n=\sum_{n\ge1}x^{n-1}=\sum_{n\ge0}x^n$

which is the geometric series. We know this series converges for $|x|$ , and this time the endpoints are not included.

This example shows that (A) is certainly possible; that is, $x=-1$ is valid in the first series, but not in the differentiated one.

- - -

Now consider the series

$\displaystyle\sum_{n\ge0}\dfrac{x^n}{n!}$

which we know to converge to $e^x$ .

Differentiating, we get

$\displaystyle\frac{\mathrm d}{\mathrm dx}\sum_{n\ge1}\frac{x^{n-1}}{(n-1)!}=\sum_{n\ge0}\frac{x^n}{n!}=e^x$

as expected. But both series converge everywhere, so this serves as a counter-example to the claim of B. So B is false.

8 0

4 years ago

Complete the proof. Someone please help me answer this!

sammy [17]

Answer:

Step-by-step explanation:

Supplementary angles are two or more angles that add up to give the measure of angle on a straight line (i.e $180^{o}$ ).

It can be deduced from the given question that;

1. <1 and <2 are supplementary, also <1 and <3 are supplementary. Reason: as given in the question.

2. Given statements: m<1 + m<2 = $180^{o}$ , and m<1 + m<3 = $180^{o}$ . The reason is the definition of supplementary angles.

3. Statement: m<2 = m<3. The reason is a transitive property.

4. Statement: m<1 + m<2 = m<1 + m<3. The reason is subtraction property of equality.

5.Given statement: <2 ≅ <3. The reason is that definition of congruent angles.

Therefore, <2 ≅ <3.

3 0

3 years ago

I am totally stuck on this problem please help

Nonamiya [84]

V solid = π * $5^{2}$ * 13 / 3 + π * $5^{2}$ * 2 = 3.14 * 25 *13 / 3 + 3.14 * 50 = 340.16 + 157 = 497.16 ≈ 497.20 cubic centimeters;

3 0

4 years ago

What is the rate of change of the linear function that passes through the points (1, 0) and (0, -3)

melisa1 [442]

Answer:

3

Step-by-step explanation:

To find the rate of change we: divide the change in y (output) by the change in x (input).

A = (-3 - 0) / (0 - 1) = -3 / -1 = 3

:)

6 0

3 years ago

Can someone please simplify this??

KIM [24]

Answer:

0

Step-by-step explanation:

=>sec A cosec A - tan A - cot A

=>(1/sinAcosA)-(sinA/cosA)-(cosA/sinA)

You get the LCM as sinAcosA then it becomes

=>(1-sin^2(A)-cos^2(A)) /sinAcosA [1-sin^2(A)=cos^2(A)]

=>(cos^2(A)-cos^2(A))/sinAcosA

=>0/sinAcosA

=>0

6 0

3 years ago