Express the series one over two add one
over four add one over six add one over eight and so on using sigma notation.
Let’s recall what sigma notation looks
like. We use the Greek letter Σ. And next to it, we have the formula for
the 𝑛th term, which I’ll call 𝑎 𝑛. Underneath the Σ, we have the first
value of 𝑛. So I’ll write 𝑛 equals 𝑘. But 𝑘 could be any number. It’s just the starting value of 𝑛. Above the Σ, we have the last value of
𝑛. Again, this could be anything; it could
even be ∞.
So we’re going to express this series
using sigma notation. So let’s start with writing Σ. We can see that the denominators here
are going up in multiples of two. So our formula for the 𝑛th term is one
over two 𝑛, starting with two times one. So underneath the Σ, we write 𝑛 equals
one. And as the series has a dot dot dot,
this implies we have infinite terms. So above the Σ, we write ∞ as that’s
technically our last value of 𝑛. So we can express this series using
sigma notation as the sum from 𝑛 equals one to ∞ of one over two 𝑛.
A useful tool which comes in handy when
dealing with infinite series is index shifts. Recall that we call 𝑛 the index of
summation. And we’ve seen that it doesn’t have to
be the letter 𝑛; it could be a different letter. And we’ve seen that the starting value
𝑘 could actually be any number. You might often see it as 𝑛 equals
zero or 𝑛 equals one, but actually it could be anything. For example, the sum from 𝑛 equals
zero to ∞ of two to the 𝑛th power, the first term when 𝑛 equals zero will be two raised
to the power of zero, which is just one. The second term, when 𝑛 equals one,
will be two raised to the first power, which is just two. And this continues in that way.
But if instead we write the sum from 𝑛
equals one to ∞ of two to the 𝑛th power, we create a different series. The first term will be two raised to
the first power because 𝑛 equals one is our starting value. So this will be two. And then it will continue on. But the important thing to note is that
changing the first value of 𝑛 gives us a different series. And we can write the same series in a
different way using index shifts.
For example, if we think about the
series the sum from 𝑘 equals one to ∞ of 𝑎𝑟 raised to the power of 𝑘 minus one. Let’s rewrite this as a series which
starts at 𝑘 equals zero. We can see that the first few terms of
this sum are 𝑎𝑟 raised to the zero power add 𝑎𝑟 raised to the first power add 𝑎𝑟
raised to the second power and so on. So we could spot that actually we could
rewrite this series as the sum of 𝑎𝑟 to the 𝑘th power starting at 𝑘 equals zero and
going to ∞. So these are exactly the same series,
but just written with a different starting value.