Series Exercises

Answer

The terms are squares, with alternating signs. The general term is

(-1)^{n + 1} n²

and we want to start at n = 2.

In summation notation the series looks like this:

For these exercises, we'll give whichever answer we think is simplest. If you get something different, consult with a teacher or friend, because you may have a different and correct answer. Like coming up with new nacho topping combinations, there are infinite ways to write the same series.

Answer

The general term is the constant 1. It doesn't matter what value of n we start on, since all the terms are the same. Both

and

are good options.

Answer

We have alternating terms, so we need a factor of (-1) raised to some power.

The general term

a_n = (-1)^{n + 1}3n

makes sense if we start at n = 1. Then the sum looks like

There are other correct answers. For example, we could change the exponent on the (-1) and write

Answer

The exponents on (.7) range from 1 to 25. We think it makes the most sense to start at n = 1 and write

Answer

The terms are even numbers starting with

8 = 2(4)

and ending with

500 = 2(250).

A reasonable way to write the sum in sigma notation is

Answer

Let's deconstruct this series. First, we can pull out a factor of 3 from each term to get

-3 + 3(0.1) – 3(0.01) + 3(0.001) – 3(.0001).

The things in the parentheses are are powers of .1, so write them that way:

-3(0.1)⁰ + 3(0.1)¹ – 3(0.1)² + 3(0.1)³ – 3(0.1)⁴

It makes sense to use n = 0 as the starting index, so to get the signs correct we need to have a factor of (-1)^{n + 1}:

(-1)¹ × 3(.1)⁰ + (-1)² × 3(.1)¹ + (-1)³ × 3(.1)² + (-1)⁴ × 3(.1)³ + (-1)⁵ × 3(.1)⁴

Finally, we put things into sigma notation:

It would also make sense to start at n = 1, which is the exponent on the first factor of -1. Then the series looks like

(-1)¹ × 3(0.1)⁰ + (-1)² × 3(0.1)¹ + (-1)³ × 3(0.1)² + (-1)⁴ × 3(0.1)³ + (-1)⁵ × 3(0.1)⁴

and we would write it in sigma notation as

Answer

This series has 16 terms, which is too many to write down reasonably. We'll do the first 3 and the last 2:

Answer

Work out the first few terms.

Sticking things together,

Answer

This is called the harmonic series. It's a very important series to remember, because it is the language of music.

Answer

This series only has 4 terms, so we might as well write them all down.

Answer

If we don't simplify the terms, we get

If we do simplify the terms, it becomes hard to see what the pattern is: