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love history [14]
3 years ago
10

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive an

y points for the skipped part, and you will not be able to come back to the skipped part.
Find a power series representation for the function. Determine the interval of convergence. (Give your power series representation centered at x=0)
f(x)=1/6+x
Mathematics
1 answer:
GaryK [48]3 years ago
4 0

Answer:

The series representation is \sum_{n=0}^{\infty}(-1)^n\frac{x^n}{6^{n+1}} and the interval of convergence is (-6,6)

Step-by-step explanation:

We want to find a series, such that f(x) = \sum_{n=0}{\infty}a_n(x-a)^{n}[/tex], were a is the value that we are using to center the series expansion. In our case, a=0.

We will use the geometric series formula as follows. For |r|<1 then

\frac{1}{1-r}=\sum_{n=0}^{\infty} r^n

In our case, with some algebreaic manipulation we have that

f(x) = \frac{1}{6+x} = \frac{1}{6}\frac{1}{1-(\frac{-x}{6})}

Taking r = \frac{-x}{6} we get that

f(x) = \frac{1}{6}\sum_{n=0}^{\infty} (\frac{-x}{6})^n = \sum_{n=0}^{\infty}(-1)^n\frac{x^n}{6^{n+1}}

This representation is valid (that means that the series converges to the value of f(x)) only for |r|<1. That is

\left|\frac{-x}{6}\right|= \left|\frac{x}{6}\right|

which implies that |x|<6. So the interval of convergence is (-6,6).

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jek_recluse [69]

Answer:

y=5^x

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Step-by-step explanation:

x=# of cycles= 15 cycles

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3 years ago
A submarine is650 feet below sea level. If it defends 400 feet, which integer represents its new position
Musya8 [376]

Answer:

-1050

Step-by-step explanation:

-650+(-400)=-1050

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You are travelling through a town at 50km/h and a ball rolls onto the road. how far do you travel (in metres) in this time.
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Answer:

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50km/h

It means that 1 hour can travelling 50 km

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4 0
3 years ago
Danny has a rectangular rose garden that measures 8m by 12.5m. One bag of fertilizer can cover 16m2 . How many bags will he need
mrs_skeptik [129]

Answer:

we will need 7 bags

Step-by-step explanation:

To solve this problem we first have to calculate the total area of ​​the garden

a = area

b = base = 8m

h = height = 12.5m

a = b * h

we replace with the known values

a = 8m * 12.5m

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Now that we know the total area of ​​the garden and the area occupied by a single bag, we divide the total area by what a bag covers.

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