Evaluate the infinite series:
[tex] \displaystyle \large{ \sum_{n = 1}^{ \infty} \frac{1}{(n + 1) \sqrt{n} + n \sqrt{n + 1} } }[/tex]
Please show your work too — thanks!
Answers
9514 1404 393
Answer:
sum = 1
Step-by-step explanation:
The method of partial sums has you look at sums of various numbers of terms of the series to see if there is a pattern that suggests either divergence or a limit. Here, the partial sums suggest the series sum is 1.
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In the attachment, we have computed sums of 1 term, 10 terms, 100 terms, and so on up to 10^8 terms. These sums seem to approach the value 1.
Individual terms get small faster than for a harmonic series, which his known not to converge, but more slowly than a quadratic series. Large numbered terms are approximately proportional to n^-(3/2).
_____
You can actually compute the sum exactly, without reference to partial sums. Each term of the sum can be rewritten as ...
[tex]\dfrac{1}{(n+1)\sqrt{n}+n\sqrt{n+1}}=\dfrac{1}{\sqrt{n(n+1)}(\sqrt{n+1}+\sqrt{n})}\\\\=\dfrac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n(n+1)}}=\dfrac{1}{\sqrt{n}}-\dfrac{1}{\sqrt{n+1}}[/tex]
Then successive terms will cancel the intermediate fractions. So, for example, the sum of the first 3 terms is ...
(1/√1 -1/√2) +(1/√2 -1/√3) +(1/√3 -1/√4) = 1 -1/√4
This means the given sum will evaluate to ...
1 -1/√∞ = 1 -0 = 1 . . . sum of the infinite series
![Evaluate The Infinite Series:[tex] \displaystyle \large{ \sum_{n = 1}^{ \infty} \frac{1}{(n + 1) \sqrt{n}](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/rpEP4lZYdhvGbef8Dwf3bhCYcpZPENPh.png)
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jamie is 3 years less than three times Andys age. In five years the sum of their ages will be 59.How old are they now
Answers
Answer:
Andy : 13 years old and
Jamie: 36 years old
Step-by-step explanation:
Andy = x
Jamie = 3x - 3
In 5 years:
(X+5)+(3x-3+5) = 59
4x + 7 = 59
4x = 59 - 7
x = 52/4
x = 13
10(c–1)=80 i need help
Answers
Answer:
C =9
Step-by-step explanation:
10(c−1)=80
Step 1: Simplify both sides of the equation.
10(c−1)=80
(10)(c)+(10)(−1)=80(Distribute)
10c+−10=80
10c−10=80
Step 2: Add 10 to both sides.
10c−10+10=80+10
10c=90
Step 3: Divide both sides by 10.
10c
10
=
90
10
c=9
Can someone help me please. Write the equation to show what would happen on the 9th term
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A water tank has the shape of a horizontal cylinder with radius 1 meter and length 2 meters. If water is being pumped into the tank at a rate of 5 I cubic meters per minute, find the rate at which the water level is rising when the water is 5 C meters deep. Simplify your answer completely.
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Step-by-step explanation:
The answer is dhdt=325πmmin.
With related rates, we need a function to relate the 2 variables, in this case it is clearly volume and height. The formula is:
V=πr2h
There is radius in the formula, but in this problem, radius is constant so it is not a variable. We can substitute the value in:
V=π(5m)2h
Since the rate in this problem is time related, we need to implicitly differentiate wrt (with respect to) time:
dVdt=(25m2)πdhdt
In the problem, we are given 3m3min which is dVdt. So we substitute this in:
dhdt=3m3min(25m2)π=325πmmin
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Answer:
y = 3[tex]\sqrt{3}[/tex]
Step-by-step explanation:
Since sinθ = opp/hypoteneuse,
sin60 = y/6
y = sin60° * 6
= [tex]\frac{\sqrt{3} }{2}[/tex] * 6
≈ 3[tex]\sqrt{3}[/tex]
Hence, y = 3[tex]\sqrt{3}[/tex]
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solve the two-step equation. -1/3x-21/4=27/4
Answers
Answer:
x=(-36)
Step-by-step explanation:
Answer:
x = -36.
Step-by-step explanation:
-1/3x-21/4=27/4
-1/3x = 27/4 + 21/4
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x = 12* -3
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It took Jack 2 hours to paddle 6.4 km upstream on the Black River to the giant's castle. The speed of the river is 1.8 km/h. How long will it take Jack to cross Swan Lake if the lake is 2.5 km long?
Answers
To find the time Jack takes to cross Swan Lake, his average speed is required:
The time it will take Jack to cross Swam Lake is 0.5 hours
Reason:
Time it took Jack to paddle 6.4 km upstream, t = 2 hours
Distance Jack paddled, d = 6.4 km
The given speed of the Black River, v₂ = 1.8 km/h
The width of the lake, w = 2.5 km
The time it will take Jack to cross Swan Lake
Let v₁, represents Jack's speed, we have;
[tex]v_1 - v_2 = \dfrac{d}{t}[/tex]
Therefore;
[tex]v_1 - 1.8 = \dfrac{6.4}{2} = 3.2[/tex]
v₁ = 3.2 + 1.8 = 5
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Your cousin’s birthday is this weekend. You want to buy a CD that costs $12.99 and a birthday card that costs $2.50. You have saved $25. Write an inequality to find the most money you have for other spending.
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Answer:
None
Step-by-step explanation:
It would be easier to ask your teacher instead of getting wrong answers from random people. Ever think of that :0
If you deposit $400 into an account that offers a 5% interest rate. How much interest will you earn after 1 year?
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Answer:
24000 grams
Hard to explain. Hope this helped! Please give brainliest.
Answer:
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Step-by-step explanation:
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Answer:
Its A
Step-by-step explanation:
took the test a while ago
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Answer:
why do you want to know
Step-by-step explanation:
hahahahahahahaha
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Answers
Answer:
the answer will be 14708
Answer:
14708
Step-by-step explanation:
Apply the formula:
Amount at time t in years =
(Starting amount)(1 + yearly per cent increase)^(Number of years) =
(13,000)(1 + 2.5%)^5 =
13000(1.025)^5 =
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Answer:
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Step-by-step explanation:
Answer:
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120 x w = 3w
W x 3 =120
3 + w = 120
120 + 3 = 3
Answers
Answer:
It would be "W x 3 = 120"
Step-by-step explanation:
Why? Well, since a swift could run 120 miles an hour, that would be a necessity in the problem. It would be needed to find out how fast the wolf ran, but, it cannot be the answer since we are trying to find out "how fast the wolf ran per hour". Right? No. Basically, you need to divide 120 by 3, which is 40. But which equals 40? None. Yes, none. No, to two answers. Why? Because you need to divide, so, the answer could be multiplication, but the opposite. You know if you know... Anyways, 120 x w = 3w is wrong due to the fact that it has no sign of division and the answer is way too large (I tried). 3 + w = 120 is definitely wrong. Why add? 3 + 40 is not 120. It's 43. Also, 120 + 3 is not 3, it's "123". So, the only answer left is "W x 3 = 120". But, how do we know that it's right? Well, think about it backwards. The swift is 3 times faster, so, the wolf times that 3 times faster is 120. And that is exactly what this problem is. By doing the problem backwards, we could find out that a wolf can run 40 miles per hour. Basically, 120 divided by 3, how much faster it is than a wolf, is 40. So, we could find out that a wolf could run 40 miles per hour.
Final Answer: W x 3 = 120
Answer:
The answer will be w×3=120
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Select the correct answer.
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O A 4y-5x= 5
OB. y - 4x = 5
OC. 5y + 4x = 5
OD. 5y - 4x = 5
Answers
Answer:
Your correct answer should be D.
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Step-by-step explanation:
Find the equation of the line that goes through (-8,11) and is perpendicular to x= - 15. Write the equation in the form x = a, y = b, or y = mx + b.
The equation is
(Type the equation using integers or fractions. Simplify your answer.)
Answers
Answer:
y=11
Step-by-step explanation:
Hi there!
We want to find the equation of the line that passes through the point (-8, 11) and is perpendicular to x=-15
If a line is perpendicular to another line, it means that the slopes of those lines are negative and reciprocal; in other words, the product of the slopes is equal to -1
The line x=-15 has an undefined slope, which we can represent as 1/0, which is also undefined.
To find the slope of the line perpendicular to x=-15, we can use this equation (m is the slope):
[tex]m_1*m_2=-1[/tex]
[tex]m_1[/tex] in this instance would be 1/0, so we can substitute it into the equation:
[tex]\frac{1}{0} *m_2=-1[/tex]
Multiply both sides by 0
[tex]m_2=0[/tex]
So the slope of the new line is 0
We can substitute it into the equation y=mx+b, where m is the slope and b is the y intercept:
y=0x+b
Now we need to find b:
Since the equation passes through the point (-8,11), we can use its values to solve for b.
Substitute -8 as x and 11 as y:
11=0(-8)+b
Multiply
11=0+b, or 11=b
So substitute into the equation:
y=0x+11
We can also write the equation as y=11
Hope this helps!
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Answers
Answer:
The Reconquista and the Inquisition are two very important events that happened in the middle ages. The Inquisition was when the Catholic church officials tortured Jews and Muslims to convert to Christianity. The Reconquista was when the people from Spain and Portugal united to take back their land from Muslims.
2x - 5y = 10
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Answer:
answer is 27 trust me. I can't explain step by step so please trust ma ans is27
I'll give brainliest if correct.
File included down below
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No solution. The first two equations would need to equal each other, but they don't, so it is mathematically impossible.
Answer: k=0
Step-by-step explanation:
[tex]kx+z=3\\\\-z -z[/tex]
[tex]kx=3-z[/tex]
[tex]\frac{kx}{k} =\frac{3}{k}-\frac{z}{k}[/tex]
[tex]x=\frac{3-z}{k}[/tex]
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Write a story that would explain the graph below.
Answers
Answer:
A kid leaping happily to home after school
The graph represents the relationship between height and time.
Given that,
A curve has been shown we have to explain the behavior of the curve.
The graph is a demonstration of curves that gives the relationship between the x and y-axis.
Here,
The curve is shown, where the vertical axis represents the height above the ground and the horizontal axis represents the time. Let's assume a plane of paper is flying, the plane getting a rise and then a certain down and again rise and down as according to the height of the plane.
Thus, the graph represents the relationship between height and time.
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Bill and Ben each have three cards, numbered 1, 2 and 3. They each take one of their own cards. Then they multiply together the two numbers on the cards. a) What is the probability that their answer is an even number? b) What is the probability that their answer is a number greater than 5?
Answers
The probabilities are given as follows:
a) Even number: 5/9.
b) Number greater than five: 1/3.
What are probabilities?A probability is the division of the number of desired outcomes by the number of total outcomes.
Given that each person draws one card from a set of three, hence the total number of outcomes is given as follows:
3² = 9.
Using the Fundamental Counting Principle,
The nine outcomes are given as follows:
1 x 1 = 1.
1 x 2 = 2.
1 x 3 = 3.
2 x 1 = 2.
2 x 2 = 4.
2 x 3 = 6.
3 x 1 = 3.
3 x 2 = 6.
3 x 3 = 9.
There are five outcomes that end with an even number, hence the probability is of:
p = 5/9.
There are three outcomes ending with a number greater than five, hence the probability is of:
p = 3/9 = 1/3.
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Answers
Step-by-step explanation:
refer to the picture.. hope this helps :)
oliver sewed a quilt 3 2/3 feet long and 2 1/4 feet wide. what is the area of her quilt?
Answers
Answer:
8.25 feet
Step-by-step explanation:
3 2/3 times 2 1/4 equals 8.25
(2 x 3) + (3 x 4) =
Answers
Answer:
18
Step-by-step explanation:
To answer, you want to follow PEMDAS:
P = parentheses
E = exponents
M = multiplication
D = division
A = addition
S = subtraction
So, the first step is parentheses, therefore we do whatever is inside the parentheses, and that is multiplying:
(2 * 3) + (3 * 4) =
6 + 12
Now, because there are no exponents, we cannot multiply anymore, and we cannot divide, we can add:
6 + 12 =
18
And that is our final answer!
Hope this helps :)