Question

ANYONE PLZ HELP ME

Answer 1

\[\sum_{x=1}^{\infty}-4(\frac{ 1 }{ 5 })^{x-1}\]

Answer 2

@wio @whpalmer4

Answer 3

Can you help me

Answer 4

Hmmm, sum of an exponential series...

Answer 5

Oh, it's a geometric series, isn't it?

Answer 6

yes

Answer 7

Isn't there something for this? Like \[
\frac{a}{1-r}
\]or something?

Answer 8

yeah i think so but I dont know how to do it

Answer 9

do you know what to do

Answer 10

Well, identify the \(a\) and \(r\) for this geometric series.

Answer 11

Do you know the formula for geometric series?

Answer 12

uhm I am not sure I have not even studied most of these formulas and they are on the quiz

Answer 13

\[
a_n = ar^{n-1}
\]

Answer 14

I have seen that before but I was not taught how to do it

Answer 15

In this case \[
a_n = -4\left(\frac 15\right)^{n-1}
\]

Answer 16

ok I think I can simplify it

Answer 17

So \[
\color{red} a\color{blue}r^{n-1} = \color{red}{-4}\left(\color{blue}{\frac{1}{5}}\right)^{n-1}
\]

Answer 18

oooh so a is -4 and r is 1/5

Answer 19

Now we use the formula \[
\sum_{n=1}^{\infty}a_n=\frac{\color{red}a}{1-\color{blue}r}
\]

Answer 20

ok so -4/(1-(1/5)) Right?

Answer 21

That's correct!

Answer 22

What do you get when you evaluate that expression?

Answer 23

cool -4/(4/5) right?

Answer 24

Keep going...

Answer 25

ok then we flip it giving us -4/1*5/4 right?

Answer 26

which =

Answer 27

-20/4=-5

Answer 28

Right?

Answer 29

for some reason, it's the kids who can do the work correctly who always want to show all the steps, and the kids who can't, and would benefit from having their mistakes identified, won't :-)

Answer 30

yes, you nailed it!

Answer 31

why don't you figure out what the first few terms of the sequence are, and see if it looks like the answer makes sense?

Answer 32

how do I find the terms using this

Answer 33

Just plug in \(x=1, 2, 3, 4, ...\)

\[x=1: -4(\frac{1}{5})^{1-1} = -4(\frac{1}{5})^0 = -4*1 = -4\]
\[x=2: -4(\frac{1}{5})^{2-1} = -4(\frac{1}{5})^1 = -\frac{4}{5}\]etc.

Answer 34

sum after the first two terms is
\[-4\frac{4}{5} = -4.8\]

Answer 35

oh ok that makes sence

Answer 36

I have a few questions having to do with this question

Answer 37

it is stuff my quiz is requesting that I dont know how to do

a. Write the first four terms of the series.
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.

Answer 38

Well, we wrote the first two terms. What are the next two? I showed you the procedure for finding them.

As for whether the series diverges or converges, if we can sum it up through infinity and get a number, doesn't that imply it converges?

Answer 39

I think I can do a but the other 2 I am not so sure

Answer 40

I'm sure your text describes the convergence test, too.

Answer 41

ok so for the 3rd term is starts with the equation

-4(1/5)^(3-1) right?

Answer 42

That then gives us -4(1/5)^2

Answer 43

-4(1/25)

Answer 44

-4/25 right?

Answer 45

@whpalmer4 you still there?

Answer 46

Sorry, didn't see the notification!
-4/25 is correct. What about the following term?

Answer 47

oh ok well it starts off as

-4(1/5)^(4-1)

Answer 48

-4(1/5)^3

Answer 49

yes. here's a shortcut — this is a geometric sequence, and the common ratio is 1/5, right? So if you have term n, you can make term n+1 by multiplying term n by 1/5...

Answer 50

Here are the first 10 terms:

\[\begin{array}{cc}
1 & -4 \\
2 & -\frac{4}{5} \\
3 & -\frac{4}{25} \\
4 & -\frac{4}{125} \\
5 & -\frac{4}{625} \\
6 & -\frac{4}{3125} \\
7 & -\frac{4}{15625} \\
8 & -\frac{4}{78125} \\
9 & -\frac{4}{390625} \\
10 & -\frac{4}{1953125} \\
\end{array}\]

Answer 51

whouh thx for the help ok so now I have the first 4 terms how do I find if the series diverges or converges

Answer 52

You were able to compute the sum of an infinite number of terms, right?

Answer 53

Would it be possible to do that if it diverges?

Answer 54

what is that again

Answer 55

If I ask you to add 1 + 2 + 3 + 4 + ... all the way up to infinity, is there an answer?

Answer 56

no

Answer 57

so it diverges right?

Answer 58

so that series 1 + 2 + 3 etc diverges.

This series we are working with adds much smaller terms as the term number gets bigger.

Answer 59

yep so since it diverges it does not have a sum right?

Answer 60

In case the fractions just make your eyes glaze over, let me show you that same table, except with decimals:

\[\begin{array}{cc}
1. & -4. \\
2. & -0.8 \\
3. & -0.16 \\
4. & -0.032 \\
5. & -0.0064 \\
6. & -0.00128 \\
7. & -0.000256 \\
8. & -0.0000512 \\
9. & -0.00001024 \\
10. & -2.048*{10}^{-6} \\
\end{array}\]

Answer 61

I like fractions better lol

Answer 62

thx so much for the help I am done with school for the day so I dont need anymore help thx so much @whpalmer4 and @wio have a good day

Answer 63

so, looking at the sum of these, does it look like term 1000 is going to make an appreciable difference? how about term 10000?

Let's look at the sum:

\[\begin{array}{c}
0 \\
-4.000000000 \\
-4.800000000 \\
-4.960000000\\
-4.992000000 \\
-4.998400000 \\
-4.999680000 \\
-4.999936000 \\
-4.999987200\\
-4.999997440 \\
\end{array}\]

Answer 64

Does it look like that is ever going to break through -5? We're adding smaller and smaller bits with each term.