So use pythagorean theorem to find the straight distance between hhouses
legs are 6 and 8
a^2+b^2=c^2
6^2+8^2=c^2
36+64=c^2
100=c^2
squrer root both sides
10=c
the distance=10 miles
Answer:
slope is 5
Step-by-step explanation:
y=mx + b
y=5x =2
m = slope therefore is 5
Answer:
Replace l with 110 and w with 120
P = 2(110) + 2(120)
P = 220 + 240
P = 460 feet
Answer:
b = -8
Step-by-step explanation:
6b = -42
Divide by 6 on both sides
b = -8
An alternating series
![\sum\limits_n(-1)^na_n](https://tex.z-dn.net/?f=%5Csum%5Climits_n%28-1%29%5Ena_n)
converges if
![|(-1)^na_n|=|a_n|](https://tex.z-dn.net/?f=%7C%28-1%29%5Ena_n%7C%3D%7Ca_n%7C)
is monotonic and
![a_n\to0](https://tex.z-dn.net/?f=a_n%5Cto0)
as
![n\to\infty](https://tex.z-dn.net/?f=n%5Cto%5Cinfty)
. Here
![a_n=\dfrac1{\ln(n+1)}](https://tex.z-dn.net/?f=a_n%3D%5Cdfrac1%7B%5Cln%28n%2B1%29%7D)
.
Let
![f(x)=\ln(x+1)](https://tex.z-dn.net/?f=f%28x%29%3D%5Cln%28x%2B1%29)
. Then
![f'(x)=\dfrac1{x+1}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cdfrac1%7Bx%2B1%7D)
, which is positive for all
![x>-1](https://tex.z-dn.net/?f=x%3E-1)
, so
![\ln(x+1)](https://tex.z-dn.net/?f=%5Cln%28x%2B1%29)
is monotonically increasing for
![x>-1](https://tex.z-dn.net/?f=x%3E-1)
. This would mean
![\dfrac1{\ln(x+1)}](https://tex.z-dn.net/?f=%5Cdfrac1%7B%5Cln%28x%2B1%29%7D)
must be a monotonically decreasing sequence over the same interval, and so must
![a_n](https://tex.z-dn.net/?f=a_n)
.
Because
![a_n](https://tex.z-dn.net/?f=a_n)
is monotonically increasing, but will still always be positive, it follows that
![a_n\to0](https://tex.z-dn.net/?f=a_n%5Cto0)
as
![n\to\infty](https://tex.z-dn.net/?f=n%5Cto%5Cinfty)
.
So,
![\sum\limits_n(-1)^na_n](https://tex.z-dn.net/?f=%5Csum%5Climits_n%28-1%29%5Ena_n)
converges.