Related questions with answers

How many times would integration by parts need to be performed to evaluate $\int x^{n} \ln x d x$ (where n is a positive integer)?

Edwina, a commodities broker, has acquired an option to buy 1,000 oz of gold at $50/oz. If she takes the option and if Congress relaxes import quotas, she can sell the gold for$80/oz. If she takes the option and Congress does not relax the import quotas, however, the company will lose $10/oz. Edwina believes that there is a 50% chance that the government will relax the quota. She also has the option of waiting until Congress decides whether to relax the import quota. If she adopts this strategy, however, there is a 70% chance that some other broker will have already taken the option. a. If Edwina is risk-neutral, what should she do? b. If Edwina’s utility function for a change x in her asset position is given by u(x) = (10,000 + x)1/2, what should she do?

Astronomers have been making maps by observing a slice of the universe and seeing where the galaxies lie within that slice. If the universe is isotropic and homogeneous, why do they need more than one slice? Suppose they now want to make each slice extend farther into the universe. What do they need to do?

Question

Why must the absolute value of the common ratio be less than 1 before an infinite geometric sequence can have a sum?

Solution

Verified

Answered 1 year ago

Notice that if a common ratio of an infinite geometric sequence is greater than $1$ , then the terms will get larger and larger which leaves a sum of infinity. However, if the common ratio is less than $1$ , the terms will get smaller and smaller which leaves a specific sum.