Math Short Answer

Clearly explain what is meant by a geometric sequence and provide a specific example.

A geometric sequence is a sequence where you get from one term to the next by multiplying by the same common ratio. For example, 2, -6, 18, -54, 162, ... is geometric because there is a common ratio of -3.

When does a geometric series converge and when does a geometric series diverge?

A geometric series converges if the absolute value of the common ratio is less than 1. A geometric series diverges if the absolute value of the common ratio is greater than or equal to 1.

Explain how a nonterminating decimal can be thought of as a series.

A nonterminating decimal is the sum of the fractions of its individual digits. More precisely, if a_n is the 𝑛th digit of the nonterminating decimal, we can express the decimal as the following sum: ∑_(n=1)^∞▒a_n/〖10〗^n .

What does it mean for a sequence to converge?

A sequence converges to the limit 𝐿 if the terms 𝑎𝑛 get closer and closer to 𝐿 as 𝑛 gets larger and larger, and we can get as close to 𝐿 as we want by choosing 𝑛 sufficiently large.

What does it mean for a series to converge?

A series converges if its sequence of partial sums converges. In other words, if there is a limit 𝐿 where the sum of the first 𝑛 terms can be made as close to 𝐿 as we want by choosing a sufficiently large number for n.

Clearly explain what is meant by an arithmetic sequence and provide a specific example.

An arithmetic sequence is a sequence where you get from one term to the next by adding the same common difference. For example, 2, 4, 6, 8, 10, ... is arithmetic because there is a common difference of 2.

Equations that have variables fall into two main categories. Give the name for the two types of equations and explain how they differ.

Equations that have variables are either conditional equations or identities. Identities are equations that at true regardless of the values of the variables, which conditional equations are true for some values of the variables and false for others.

If the nth term of a sequence can be given by a rational expression, under what conditions will the sequence converge and what will the limit be? Under what conditions with the sequence diverge?

If the degree of the numerator is less than the degree of the denominator, then the sequence converges to 0. If the degree of the numerator and the degree of the denominator are equal, then the sequence converges to the ratio of the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, then the sequence diverges.

Explain how the limit/convergence concept is used in derivatives and integrals.

Recall that the concept of limit/convergence is that a(n) (infinite) sequence converges if the terms get arbitrarily close to a particular number the further down the list we go. For the derivative, we take the slope between a particular point we're interested in and nearby points. As we let the nearby point get closer and closer to the particular point, if the sequence of slopes has a limit, then that is what we call the derivative at the particular point. For the integral, we approximate the area under the curve by breaking up the area into rectangles. The more rectangles we fit into the same overall width, the closer our approximation gets to the exact area. When we take the sequence of area estimates as the number of rectangles increases without bound, if the limit of this sequence converges, then that is what we call the exact area under the curve, or also the definite integral of the function.

Explain why it is impossible to have a slope through a single point, yet we can find the slope of a function through a single point on its graph.

The formula for slope is (y_2-y_1)/(x_2-x_1 ), which evaluates to 0/0 and is undefined if we use the same point (x_1,y_1 ) and (x_2,y_2). Thus we can't use just one point to compute a slope. However, with limits, we can take the slope between the point of interest and neighboring points on the graph and look at the slope between these points. We can let the neighboring point get closer and closer to the point of interest. If this sequence of slopes converges as the points get closer and closer together, then that is what we will say the slope at the single point will be.

Why is the real number 0.9999 exactly equal to the real number 1?

The nonterminating decimal 0.99999... is the sum of the geometric series with first term 0.9 and common ratio 0.1. Thus this sum is equal to 0.9/(t-0.1)=1. Thus the nonterminating decimal 0.9999... represents the same real number as 1. Thus they are exactly equal.

Suppose you are given a function f(x) and that the real number L is the limit of f(x) as x approaches the real number a. In a sentence or two, informally but precisely explain the concept of limit in this situation.

The number L is the limit of f(x) as x approaches a means that we consider x-values near a but not exactly equal to a, ask the question "As we plug in x-values closer and closer to a, is there one particular real number that the corresponding y-values get closer and closer to?", and we get the answer "L." More precisely, we can find points whose y-coordinates as close to L as we want by plugging in x-values close enough to a. The actual function value f(a) is not relevant to the calculation unless the function is continuous at a. In that case, the limit is equal to the function value. We say that L is the limit if, when we consider x-values that get arbitrarily close to a, the corresponding y-vaules get arbitrarily close to L. In other words, when we plug in values of x that are close to a, the y-values we get as outputs are close to L, and we can get as close to L as we want by picking x-coordinates close enough to a.

Are there any sequences that are both arithmetic and geometric? If so, give such a sequence. If not, briefly explain why no such sequence can exist.

Yes, 0,0,0,0,0,0,0.