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lora16 [44]
3 years ago
10

By recognizing each series below as a Taylor series evaluated at a particular value of x, find the sum of each convergent series

.
A. 1 + 1/5 + (1/5)^2 + (1/5)^3 + (1/5)^4 +.....+ (1/5)^n + .... = _____.
B. 1 + 5 + 5^2/2! + 5^3/3! + 5^4/4! +....+ 5^n/n! +....= _____.
Mathematics
1 answer:
goldfiish [28.3K]3 years ago
4 0

The first sum is a geometric series:

1+\dfrac15+\dfrac1{5^2}+\dfrac1{5^3}+\cdots+\dfrac1{5^n}+\cdots=\displaystyle\sum_{n=0}^\infty\frac1{5^n}

Recall that for |<em>x</em>| < 1, we have

\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n

Here we have |<em>x</em>| = |1/5| = 1/5 < 1, so the first sum converges to 1/(1 - 1/5) = 5/4.

The second sum is exponential:

1+5+\dfrac{5^2}{2!}+\dfrac{5^3}{3!}+\cdots+\dfrac{5^n}{n!}+\cdots=\displaystyle\sum_{n=0}^\infty \frac{5^n}{n!}

Recall that

\exp(x)=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}

which converges everywhere, so the second sum converges to exp(5) or <em>e</em>⁵.

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A pet shop sells goldfish for $ 9 per dozen. How much would 6 goldfish cost? How many goldfish could you buy with $50?
deff fn [24]

Answer:

6 goldfish = $4.50. You can buy 66 goldfish with $50

Step-by-step explanation:

We know that dozen is 12, so 12 goldfish would be $9. To find how much one goldfish costs we would divide 9 and 12. 9÷12 is 0.75. So one goldfish is $0.75.

To find 6 goldfish just multiply 0.75 to 6. 6 goldfish would be $4.50.

We know that 0.75 times 100 would be 75 so you can only buy less than 100 goldfishes. You can buy 66 goldfishes with $50.

If this helped, I would appreciate it if you marked me brainliest. Have a nice day!

6 0
3 years ago
A wind turbine has three equally spaced blades that are each 131 feet long. What is the measure of the arc between any two blade
bogdanovich [222]

Answer:

274.23 feet

Step-by-step explanation:

From the question,

The measure of the arc between any two blades (s) = 2πrΦ/360

s =  2πrΦ/360................... Equation 1

Where Ф = angle formed betwen two blades, r = length of each blade.

Given: Ф = 360/3 (the three blades are equally spaced) = 120°, r = 131 feet.

Constant: π = 3.14

Substitute these values into equation 1

s = (2×3.14×131×120)/360

s = 274.23 feet.

Hence, the measure of the arc between any two blades is given = 274.23 feet

4 0
3 years ago
WILL MARK BRAINLIEST - 75 POINTS!!
Tpy6a [65]

Answer:

(10x-1)^2

Step-by-step explanation:

Here we are going to apply the rule of "Square of the difference". The rule is as given below

(a-b)^2=a^2-2 \times a \times b + b^2   ---------(A)

Now we have our original polynomial as

100x^2-20x+1

Which can be re written as

(10x)^2-2 \times 10x \times 1 + 1^2  ---------------(B)

Hence comparing it with (A)

a=10x

b=1

Applying the rule on (B)

(10x)^{2}-2 \times 10x \times 1 + 1^2 = (10x-1)^2

Hence our answer is

(10x-1)^2

6 0
3 years ago
Read 2 more answers
(2<img src="https://tex.z-dn.net/?f=%5Csqrt%7B7" id="TexFormula1" title="\sqrt{7" alt="\sqrt{7" align="absmiddle" class="latex-f
evablogger [386]
Answer is: 1
I used a lot of different methods & it always equal one.

3 0
3 years ago
PLEASE HELP WILL GIVE YOU ALL MY POINTS In the picture, t || s, mz1 = 10x + 2, and mz2 = 12x - 22. Find the measure of angle 2.
andre [41]

Answer:

\huge\boxed{\sf < 2 = 122\°}

Step-by-step explanation:

From the figure,

<h3>∠1 = ∠2 </h3>
  • (Alternate angles are equal)

<u>We know that:</u>

  • ∠1 = 10x + 2
  • ∠2 = 12x - 22

So,

<h3>10x + 2 = 12x - 22</h3>

Add 22 to both sides

10x + 2 + 22 = 12x

10x + 24 = 12x

Subtract 10x to both sides

24 = 12x - 10x

24 = 2x

Divide 2 to both sides

12 = x

OR

<h3>x = 12</h3>

Given that,

∠2 = 12x - 22

Put x = 12

∠2 = 12(12) - 22

∠2 = 144 - 22

<h3>∠2 = 122°</h3>

\rule[225]{225}{2}

8 0
2 years ago
Read 2 more answers
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