Answer:
A. To find the partial sums of the series ∑n/(n+1)! from n = 1 to n = 4, we plug in the values of n and add them up:
s1 = 1/2! = 1/2
s2 = 1/2! + 2/3! = 1/2 + 2/6 = 2/3
s3 = 1/2! + 2/3! + 3/4! = 1/2 + 2/6 + 3/24 = 11/12
s4 = 1/2! + 2/3! + 3/4! + 4/5! = 1/2 + 2/6 + 3/24 + 4/120 = 23/30
The denominators of the terms in the partial sums are the factorials, specifically (n+1)!.
We notice that the terms in the numerator of the series are consecutive integers starting from 1. Therefore, we can write the nth term as n/(n+1)!, which can be expressed as (n+1)/(n+1)!, or simply 1/n! - 1/(n+1)!. Thus, the series can be written as:
∑n/(n+1)! = ∑[1/n! - 1/(n+1)!]
Using this expression, we can write the partial sum sn as:
sn = 1/1! - 1/(2!) + 1/2! - 1/(3!) + 1/3! - ... + 1/n! - 1/((n+1)!)
B. To prove that the formula for sn is correct, we can use mathematical induction.
Base case: n = 1
s1 = 1/1! - 1/(2!) = 1/2, which matches the formula for s1.
Inductive hypothesis: Assume that the formula for sn is correct for some value k, that is,
sk = 1/1! - 1/(2!) + 1/2! - 1/(3!) + 1/3! - ... + 1/k! - 1/((k+1)!).
Inductive step: We need to show that the formula is also correct for n = k+1, that is,
sk+1 = 1/1! - 1/(2!) + 1/2! - 1/(3!) + 1/3! - ... + 1/k! - 1/((k+1)!) + 1/((k+1)!) - 1/((k+2)!).
Simplifying this expression, we get:
sk+1 = sk + 1/((k+1)!) - 1/((k+2)!)
Using the inductive hypothesis, we substitute the formula for sk and simplify:
sk+1 = 1/1! - 1/(2!) + 1/2! - 1/(3!) + 1/3! - ... + 1/k! - 1/((k+1)!) + 1/((k+1)!) - 1/((k+2)!)
= 1/1! - 1/(2!) + 1/2! - 1/(3!) + 1/3! - ... + 1/k! + 1/((k+1)!) - 1/((k+2)!)
= ∑[1/n! - 1/(n
By examining the first few terms, we can see that the denominators are factorial expressions with a shift of 1, i.e., (n+1)! = (n+1)n!. Using this pattern, we can guess that the nth partial sum of the series is given by sn = 1 - 1/(n+1).
The given series is a sum of terms of the form n/(n+1)! which have a pattern in their denominators.
To prove this guess, we can use mathematical induction. First, we note that s1 = 1 - 1/2 = 1/2. Now, assuming that sn = 1 - 1/(n+1), we can find sn+1 as follows:
sn+1 = sn + (n+1)/(n+2)!
= 1 - 1/(n+1) + (n+1)/(n+2)!
= 1 - 1/(n+2).
This confirms our guess that sn = 1 - 1/(n+1).
To show that the series is convergent, we can use the ratio test. The ratio of consecutive terms is given by (n+1)/(n+2), which approaches 1 as n approaches infinity. Since the limit of the ratio is less than 1, the series converges. To find its sum, we can use the formula for a convergent geometric series:
∑ n/(n+1)! = lim n→∞ sn = lim n→∞ (1 - 1/(n+1)) = 1.
Therefore, the sum of the given infinite series is 1.
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Can anyone help me with this please?
Answer: 3/1
Step-by-step explanation:
use the 2 bottom ones
prove that the polynomials are nonnegative a^2-2ab+b^2+c^2
Answer:
a2 + b2 + c2 - ab - bc - ca
= (1/2)(2a2 + 2b2 + 2c2 - 2ab - 2bc - 2ca)
= (1/2)(a2 - 2ab + b2 + b2 - 2bc + c2 + c2 - 2ca + a2)
= (1/2)[(a-b)2 + (b-c)2 + (c-a)2]
Now we know that, the square of a number is always greater than or equal to zero.
Hence, (1/2)[(a-b)2 + (b-c)2 + (c-a)2] ≥ 0 => it is always non-negative.
A blood bank needs 6 people to help with a blood drive. 13 people have volunteered. Find how many different groups of 6 can be formed from the 13 volunteers.
The number of different groups of 6 that can be formed from the 13 volunteers is; 1716
How to solve permutation and combination?
This is a combination question that follows that formula;
nCr = n!/(r!(n - r)!)
We are told that the blood bank needs 6 people to help with a blood drive and that 13 have already volunteered.
Thus, number of different groups of 6 that can be formed from the 13 volunteers is;
¹³C₆ = 1716
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Phil is buying a sandwich. He has the following choices.
Bread: wheat (W), italian (I), baguette (B)
Cheese: grated (G), sliced (S)
Meat: chicken (C), turkey (T), fish (F), ham (H)
If each sandwich consists of one type of bread, one type of cheese, and one type of meat, which best represents four of his possible sandwich choices?
WIG, WCT, WGS, WST
WG, WS, WC, WT
WGT, WGF, WSC, WSH
WGSC, WGT, WBS, WGSF
Answer:
its WG, WS, WC, WT
Step-by-step explanation:
Answer:
wgt, wgf, wsc, wsh
Step-by-step explanation:
How do I solve this?
Answer:
sin (330)
csc (60)
cos (390)
Step-by-step explanation:
Solve the trigonometric equation by isolating the function and then taking the inverse. Use the period to find the full set of all solutions
Khushali selects three different numbers from the set
{−7, −5, −3, −1, 0, 2, 4, 6, 8}. She then finds the product of the three chosen
numbers. What is the largest product that Khushali can make?
Answer:
The sample space for given condition is
(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)
(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)
(1,6),(2,6),(3,6),(4,6),(5,6)
(1,5),(2,5),(3,5),(4,5)
(1,4),(2,4),(3,4)
(1,3),(2,3)
(1,2)
Number of sample space is 28.
Number of sample space where the sum of two numbers is more than 13 is 2((6,8),(7,8)). Therefore the probability is
28
2
=
14
1
Step-by-step explanation:
please thank to all my answer and please mark as brilliant
65/100 as a dollar amount
Answer:
The answer is $0.65, or 65 cents.
Answer:
$.65
Step-by-step explanation:
so 65 cents
how to solve 15x^3/4
Answer:
3
Simplify —
4
Equation at the end of step
1
:
3
15x • —
4
STEP
2
:
Final result :
45x
———
4
Step-by-step explanation:
Answer:
Well...without the x I know it is 843.75 but I don't know the fraction of it so sorry! I try my best to help people!
Please answer! I will give brainliest, it would make my day :-)
Write a formula for the given measure. Let P represent the perimeter in inches, and w represent width in inches. Identify which variable depends on which in the formula.
the perimeter of a rectangle with a length of 5 inches
P=
Step-by-step explanation:
p = perimeter
w = width
let the length be l
as we know that
perimeter of rectangle (p).= 2(l+ w)
p = 2 ( 5+w)
p = 10 + 2w
so here variable p is dependent on the value of w.
hope this will be helpful to you .plz mark my answer as brainlist plzzzz if you find it useful.☺️.
What is the result when 3x3 + 20x2 – 11x – 28 is divided by x + 7?
Answer:
Step-by-step explanation:
Let's perform this division using synthetic division. The coefficients of the given expression are {3, 20, -11, -28}, and the appropriate divisor is -7 (that comes from x + 7):
-7 / 3 20 -11 -28
-21 7 28
-------------------------------------
3 -1 -4 0
Since the remainder is 0, -7 is a root of the given expression, and the coefficients of the quotient are {3, -1, -4}. The quotient is therefore
3x^2 - x - 4
7/20% off 25 is what?
Answer:
As a fraction: 7/80
0.0875
Step-by-step explanation:
If you meant 20% it is 5.
You can work a total of no more than 35 hours each week at your two jobs. Housecleaning pays $7 per hour and your sales job pays $9 per hour. You need to earn at least $314 each week to pay your bills. Write a system of inequalities that shows the various numbers of hours you can work at each job.
Answer:
Look Below
Step-by-step explanation:
7x+9x=314 I think that is what you are looking for I am not sure. Please do not be mad if wrong.
The system of equation of inequalities in various numbers of hours are
h+s≤35
7h + 9s >_314.
To write a system of inequalities.
What is inequality?The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than’, or < ‘less than’.
Given that:
The system of equation of inequalities are
h+s≤35
7h + 9s >_314
Therefore , the system of equation of inequalities in various numbers of hours are
h+s≤35
7h + 9s >_314.
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f(n) = 2• (-3)^n
complete the recursive formula of f(n).
f(1) = ____
f(n) = f(n - 1) • ____
Answer:
[tex]f(1)=-6[/tex]
[tex]f(n)=f(n-1)(-3)[/tex]
Step-by-step explanation:
We are given that
[tex]f(n)=2\cdot (-3)^n[/tex]
We have to complete the recursive formula of f(n).
Substitute n=1
[tex]f(1)=2\cdot (-3)[/tex]
[tex]f(1)=-6[/tex]
[tex]f(2)=2\cdot (-3)^2=18[/tex]
[tex]f(3)=2\cdot (-3)^3=-54[/tex]
[tex]\frac{f(2)}{f(1)}=\frac{18}{-6}=-3[/tex]
[tex]\frac{f(3)}{f(2)}=\frac{-54}{18}=-3[/tex]
It forms geometric sequence because the ratio of two consecutive terms are equal.
Therefore, the recursive formula
[tex]f(n)=f(n-1)r[/tex]
[tex]f(n)=f(n-1)(-3)[/tex]
Answer:
f(1) = -6
f(n)= f(n−1)⋅ -3
Step-by-step explanation:
The point (-7,3) is on the line given by which of the equations below?
A. y = 3x
B. y= 7x
C. y= x + 10
D. y= x-10
what equation?
I need help with math if you don't know the answer get out of the question and don't answer it and give the wrong answer or some link or even say "I don't know" or anything like that it's not fair to me cause it a waste of my points I will report your if you do that :) have a nice day and stay safe
Answer:
Please do not report me if this is wrong but my friend beside me is giving me the answer and she said its 60
Step-by-step explanation:
which expression is equivalent to (x ^4) ^-5
Answer:1/x^20
Step-by-step explanation:
Answer:
[tex]\frac{1}{x^{20}}[/tex]
Step-by-step explanation:
[tex](x^{4})^{-5}[/tex] = [tex]\frac{1}{(x^{4})^{5}}[/tex] = [tex]\frac{1}{x^{20}}[/tex]
Given the histogram, use the drop-down menus to identify the elements.
Zero frequency: (300-350) (351-400) (401-450
A peak: (650-700) (601-650) (451-500)
the ones in parenthisis r the answer choices
Answer:
Zero frequency:
✔ 351–400
A peak:
✔ 651–700
Step-by-step explanation:
Answer:
zero frequency
351-400
A peak
651-700
Step-by-step explanation:
Yo i need help refer to picture
Step-by-step explanation:
AA: Two pairs of corresponding angles are equal.
ASA (Angle-Side-Angle)ASA: Two pairs of corresponding angles and the corresponding sides between them are equal. hope this helps you. means similarity would be equal.. I cannot read your paragraphs that well they are too small but I hope this helps you just got to find which paragraph it shows that they AA and AAS a are similar to each other by being equal
8 x ÷ 4 = 100 What is the value of x in the equation?
5
A. 5
B. 25
C. 50
D. 200
Answer:
Hi there. I had put a comment replying to your question as I didn't see any place that I could answer. If you still need it, then the answer was
C. 50
Step-by-step explanation:
8 * 50 = 400
400 / 4 = 100
In any way possible, I hope this helps you! Have a great day!
Which linear function represents the table
PLEASE HELP ME ILL GIVE BRAINLIEST
Answer:
the answer is B y=6x+2
Step-by-step explanation:
Answer:
The answer is B.
Step-by-step explanation:
Hope this helps.
Help me for a cookie
Answer:
49
Step-by-step explanation:
SUBTRACTION IDIOT
Answer:
X= 49
Step-by-step explanation:
Right angle= 90 degrees = subtraction (90-41)
Find the slope between (1,-19) and (-2,7)
Answer:
m= -26/3
Step-by-step explanation:
I used an app called "M a t h w a y " (with no spaces) you should use it
it is very helpful!
can someone please help me again, i suck at math..
Answer:
18.55 - 1.75= 16.8
16.8/12= 1.4 oz
Answer- 1.4 oz
Step-by-step explanation:
Sorry if this is wrong, I suck at math, too.
The graph shows two lines, A and B.
у
6
A
B
5
4
3
2
1
-X
0
1
2
3
4
5
6
Part A: How many solutions does the pair of equations for lines A and B have? Explain your answer. (5
Part B: What is the solution to the equations of lines A and B? Explain your answer. (5 points)
helpppppppppppppppppppppppppppppppp plzzzzzzzzzzzzzzzzzzzz
Which is the best first step when solving the following system of equations?
5C+40= 10
3C+2d=16
Answer:
multiply the second equation by - 2
Step-by-step explanation:
Given the 2 equations
5c + 4d = 10 → (1)
3c + 2d = 16 → (2)
Multiplying (2) by - 2 and adding to (1) will eliminate d
- 6c - 4d = - 32 → (3)
Add (1) and (3) term by term to eliminate d
- c + 0 = - 22
- c = - 22 ( multiply both sides by - 1 )
c = 22
Substitute c = 22 into either of the 2 equations and solve for d
Substituting into (2)
3(22) + 2d = 16
66 + 2d = 16 ( subtract 66 from both sides )
2d = - 50 ( divide both sides by 2 )
d = - 25
solution is c = 22, d = - 25
C = -y - a/b
Solve the variable a
Can you please show each step
Find the length of the third side. If necessary, write in simplest radical form. 1 2
Answer:
√3
Step-by-step explanation:
Using the pythagorean theorem, we know that a^2 + b^2 = c^2
In this case, c = 2, which means that a^2 + b^2 = 4
Since we know that a = 1, this means that a^2 is also 1. So, we know that b^2 = 4 - 1
Therefore, b^2 = 3
So, the answer is square root 3
Plz mark as brainliest :D!
Joe borrowed $4,000 from the bank at a rate of 7% simple interest per year. How much interest did he pay in 3 years?
Answer:
Wmkwkqkw siwiq q wksos ww wkwlwkw wlw
Answer:
$840
Step-by-step explanation:
4000 x .07= 280
280 x 3 = 840
please do them all please
Answer:
7(3 + 3x) = 21x + 21
2d + 10(4d +4 + 2a) = 20a + 42d + 40
3t + 2x + x = 3t + 3x
5(r + 12) - 25 = 5r + 35
18s + 12rs + 10r - 2rs = 10rs + 10r + 18s
s(12 - 4) + 2 = 8s + 2
w(15 - 3) + 2 + 7w = 19w + 2
Step-by-step explanation: