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I need help with this problem from the calculus portion on my ACT prep guide

LenaWriter [7]

Given a series, the ratio test implies finding the following limit:

$\lim _{n\to\infty}\lvert\frac{a_{n+1}}{a_n}\rvert=r$

If r<1 then the series converges, if r>1 the series diverges and if r=1 the test is inconclusive and we can't assure if the series converges or diverges. So let's see the terms in this limit:

$\begin{gathered} a_n=\frac{2^n}{n5^{n+1}} \\ a_{n+1}=\frac{2^{n+1}}{(n+1)5^{n+2}} \end{gathered}$

Then the limit is:

$\lim _{n\to\infty}\lvert\frac{a_{n+1}}{a_n}\rvert=\lim _{n\to\infty}\lvert\frac{n5^{n+1}}{2^n}\cdot\frac{2^{n+1}}{\mleft(n+1\mright)5^{n+2}}\rvert=\lim _{n\to\infty}\lvert\frac{2^{n+1}}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^{n+1}}{5^{n+2}}\rvert$

We can simplify the expressions inside the absolute value:

$\begin{gathered} \lim _{n\to\infty}\lvert\frac{2^{n+1}}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^{n+1}}{5^{n+2}}\rvert=\lim _{n\to\infty}\lvert\frac{2^n\cdot2}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^n\cdot5}{5^n\cdot5\cdot5}\rvert \\ \lim _{n\to\infty}\lvert\frac{2^n\cdot2}{2^n}\cdot\frac{n}{n+1}\cdot\frac{5^n\cdot5}{5^n\cdot5\cdot5}\rvert=\lim _{n\to\infty}\lvert2\cdot\frac{n}{n+1}\cdot\frac{1}{5}\rvert \\ \lim _{n\to\infty}\lvert2\cdot\frac{n}{n+1}\cdot\frac{1}{5}\rvert=\lim _{n\to\infty}\lvert\frac{2}{5}\cdot\frac{n}{n+1}\rvert \end{gathered}$

Since none of the terms inside the absolute value can be negative we can write this with out it:

$\lim _{n\to\infty}\lvert\frac{2}{5}\cdot\frac{n}{n+1}\rvert=\lim _{n\to\infty}\frac{2}{5}\cdot\frac{n}{n+1}$

Now let's re-writte n/(n+1):

$\frac{n}{n+1}=\frac{n}{n\cdot(1+\frac{1}{n})}=\frac{1}{1+\frac{1}{n}}$

Then the limit we have to find is:

$\lim _{n\to\infty}\frac{2}{5}\cdot\frac{n}{n+1}=\lim _{n\to\infty}\frac{2}{5}\cdot\frac{1}{1+\frac{1}{n}}$

Note that the limit of 1/n when n tends to infinite is 0 so we get:

$\lim _{n\to\infty}\frac{2}{5}\cdot\frac{1}{1+\frac{1}{n}}=\frac{2}{5}\cdot\frac{1}{1+0}=\frac{2}{5}=0.4$

So from the test ratio r=0.4 and the series converges. Then the answer is the second option.

8 0

2 years ago

Which functions have an additive rate of change of 3? Select two options.

mezya [45]

The answers would be 9 and 6

8 0

3 years ago

timmy is 3 feet four inches tall. there are 2.54 centimeters in one foot what is timmys hight on centimeters

svp [43]

Answer:

101.6 centimeters

Step-by-step explanation:

l

4 0

3 years ago

I WILL GIVE BRAINLIEST!! PLEASE HELPPP! The graph shows two lines, A and B:

gtnhenbr [62]

Answer:

(3, 4) is the solution to both lines A and B.

Step-by-step explanation:

First find the slope of each line by using the slope formula

Slope of A: -2 (passes through (2,6) and (5,0))

Slope of B: 2/3 (passes through (0,2) and (6,6))

Next, find the value of b by plugging each slope value and each corresponding x and y into the point slope equation y=mx+b

Slope A:

(6)=-2(2)+b I plugged in the point (2,6) but you can also use (5,0).

6 = -4+b Next solve for b

10=b, The equation of the line that passes through (2,6) and (5,0) is y=-2x+10

Slope B:

(2)=2/3(0)+b I plugged in the point (0,2) but you can also use (6,6).

2=b, The equation of the line that passes through (0,2) and (6,6) is

y=2/3x+2

Now you can solve for the system of equations using either substitution or elimination. I will use substitution to first find the value of x.

y=-2x+10

y=2/3x+2

-2x+10=2/3x+2 Solve for x.

8=2 2/3x

x=3

Next plug in x to either of the equations to solve for y.

y=-2(3)+10

y=4, The solution to the system of equations is (3,4)

To check, plug (3,4) into the equation y=2/3x+2

(4)=2/3(3)+2

4=2+2

4=4 This works therefore, (3, 4) is the solution to both lines A and B.

3 0

3 years ago

Which polynomial is in standard form? 12x-14x^4+11x^5

Karo-lina-s [1.5K]

3 0

3 years ago