Evaluate the geometric series or state that it diverges.

[infinity]
Σ e^-5n
n=0

sum = = 1 / ( 1-e^(-5) )

( 1.00678   to 5 decimals)

Step-by-step explanation:

Question: sum

[infinity]

Σ e^-5n

n=0

We note that this is a geometric series,

with n=0, e^(-5n) = e^0 = 1

with n=1, e^(-5n) = e^(-5)

with n=2, e^(-5n) = e^(-10)

...

Therefore the common ratio is r=e^(-5)

and the sum for a geometric series

a+ar+ar^2 + ... ar^N

=a(1-r^(N+1)) / (1-r)

=1(1-r^(infinity-1)) / (1-e^(-5))

= 1 / ( 1-e^(-5) )

x = 10.1

step-by-step explanation:

all of the angles of a tirangle add up to 180º

180 - 65 = 115 (since we already know the value of one angle)

(2x + 7) + (8x + 7) = 115

(combins like terms) 10x + 14 = 115

(isolate the variable subtract 14 from both sides): 10x = 101

(find the variable divide both sides by 10): x = 101/10

x = 10.1

(sin 42)/31. = sinx)/37

sinx = 37 (sin42)/31

= 37× 0.66/31 = 0.787

x= 51.9

other angle opposite to y = 180-51.9-42

= 180- 93.9 =86.1

so

sin42)/31 = sin 86.1)/y

0.021 = 0.997 /y

y= 0.997/0.021 =47.47