What is 5/4 of a right angle?

(a) Suppose anxn has finite radius of convergence R and an ≥ 0 for all n. Show that if the series converges at R, then it also c

valina [46]

Answer:

a) See the proof below.

b) $\sum \frac{(-x)^n}{n}$

Step-by-step explanation:

Part a

For this case we assume that we have the following series $\sum a)n x^n$ and this series has a finite radius of convergence $R$ and we assume that $a_n \geq 0$ for all n, this information is given by the problem.

We assume that the series converges at the point $x= R$ since w eknwo that converges, and since converges we can conclude that:

$\sum a)n R^n < \infty$

For this case we need to show that converges also for $x=-R$

So we need to proof that $\sum a_n (-R)^n < \infty$

We can do some algebra and we can rewrite the following expression like this:

$\sum a_n (-R)^n = \sum (-1)^n a)n R^n$ and we see that the last series is alternating.

Since we know that $\sum a_n x^n$ converges then the sequence { $a_n R^n$ } must be positive and we need to have $lim_{n\to \infty} a^n R^n = 0$

And then by the alternating series test we can conclude that $\sum a_n (-R)^n$ also converges. And then we conclude that the power series $a_n x^n$ converges for $x=-R$ ,and that complete the proof.

Part b

For this case we need to provide a series whose interval of convergence is exactly (-1,1]

And the best function for this $\frac{(-x)^n}{n}$

Because the series $\sum \frac{(-x)^n}{n}$ converges to $-ln(1+x)$ when $|x|$ using the root test.

But by the properties of the natural log the series diverges at $x=-1$ because $\sum \frac{1}{n} =\infty$ and for $x=1$ we know that converges since $\sum \frac{-1}{n}$ is an alternating series that converges because the expression tends to 0.

6 0

3 years ago

The distance a butterfly flies varies directly with it's time in the air. If a butterfly travels 264 ft in 0.5 min, how far will

Eva8 [605]

Answer:

The butterfly travels 660 ft in 1.25 min.

Step-by-step explanation:

If the distance, d, a butterfly flies varies directly as the time,t in the air, then we can write the mathematical statement.

$d\propto t$

If we introduce the constant of proportionality, we obtain;

$d=kt$

According to the question, when $d=264\:ft$ , $t=0.5\:min$ .

We substitute these values to obtain;

$264=0.5k$

$\Rightarrow k=\frac{264}{0.5}$

$\Rightarrow k=528$

The direct variation equation now becomes

$d=528t$

When $t=1.25\:min$

$d=528(1.25)$

$d=660$

Therefore the butterfly will travel 660 ft in 1.25 min.

7 0

3 years ago

What percentage is the shaded area

Sati [7]

90%, as each section is 10 percent!

7 0

3 years ago

Read 2 more answers

PLEASE HELP ME! I really need help y'all...

ikadub [295]

just a quick clarification, tis usually -4.9 and that's a rounded number to reflect earth's gravity on an object in motion, but -5 is close enough :)

$\bf ~~~~~~\textit{initial velocity in meters} \\\\ h(t) = -4.9t^2+v_ot+h_o \quad \begin{cases} v_o=\textit{initial velocity}\\ \qquad \textit{of the object}\\ h_o=\textit{initial height}\\ \qquad \textit{of the object}\\ h=\textit{object's height}\\ \qquad \textit{at "t" seconds} \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf h(x)=-5(\stackrel{\mathbb{F~O~I~L}}{x^2-8x+16})+180\implies h(x)=-5x^2+40x-80+180 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill h(x)=-5x^2+\stackrel{\stackrel{v_o}{\downarrow }}{40} x+\stackrel{\stackrel{h_o}{\downarrow }}{\boxed{100}}~\hfill$