He following series are geometric series or a sum of two geometric series. determine whether each series converges or not. for the series which converge, enter the sum of the series. for the series which diverges enter "div" (without quotes). (a) ∑n=1∞8n7n= , (b) ∑n=2∞12n= , (c) ∑n=0∞2n42n+1= , (d) ∑n=5∞7n8n= 7/8 , (e) ∑n=1∞8n8n+4= , (f) ∑n=1∞7n+2n8n=

An automated water dispenser fills packets with one liter of water on average, with a standard deviation of 5 milliliter. the manual says that after a year of operation the dispenser should be tested to see if it needs recalibration. a year later a number of filled packets are set aside and measured separately. it is found that the average packet now contains about 0.995 liters. does the dispenser need calibration? explain your answer.

Rectangle leg this 11feet and perimeter of 38feet what is the width of the rectangle in feet

Is the product of two rational numbers irrational or rational? first, make a hypothesis by multiplying two rational numbers. then, use variables such as x=a/b and y=c/d and the closure property of integers to prove your hypothesis.