Video Transcript
Express the series the sum from 𝑛
equals one to ∞ of 𝑛 add one over 𝑛 to the fourth power as a series that starts at 𝑛
equals three.
Let’s recall what this sigma notation
means. 𝑛 equals one is our starting value for
𝑛. And 𝑛 is our index of summation. The value above the Σ gives us our last
value of 𝑛. And next to the Σ, we have the formula
to find the 𝑛th term. But we want to start this series at a
different value for 𝑛. But we don’t want to actually change
the value of the series itself. If we simply change 𝑛 equals one to 𝑛
equals three, we would end up losing two terms, which would then change the value of the
series. So instead, we perform an index
shift.
If we start by defining a variable 𝑖
equals 𝑛 add two so that when 𝑛 equals one, 𝑖 equals three, which is what we want our
new starting value to be. We can also replace the value above the
Σ, the last value of 𝑛. So when 𝑛 equals ∞, 𝑖 equals ∞ plus
two. But this is just ∞. So when we rewrite our series using 𝑖
equals 𝑛 add two, the value above the Σ ,∞, will stay the same. So as we’re changing our starting value
for the series, we will end up changing the formula for the 𝑛th term. And we do this by replacing 𝑛 using
the new variable that we defined.
Note that if 𝑖 equals 𝑛 add two, then
we can say that 𝑛 equals 𝑖 minus two just by rearranging for 𝑛. So now, we’re going to rewrite our
series in terms of 𝑖. So we replace 𝑛 with 𝑖. From here, we can just simplify a
little bit. On the numerator here, we have 𝑖 minus
two add one. So this is just 𝑖 minus one. And the denominator stays the same. Now, remember that, with sigma
notation, it doesn’t matter what letter we use. So let’s change the 𝑖 to an 𝑛. So this gives us the sum from 𝑛 equals
three to ∞ of 𝑛 minus one over 𝑛 minus two raised to the fourth power.
If we take the series we were given in
the question and find the first few terms and then simplify them, we can compare it with
the series that we found for our answer. And then again, we can simplify the
terms to check that our answer gives the same series as the series we were given in the
question.
Another way that we can write the same
series in a different way is to remove terms from the summation. Let’s say we have the series the sum
from 𝑛 equals one to ∞ of 𝑎 𝑛. Then the first term is 𝑎 one and the
second term is 𝑎 two and the third term is 𝑎 three and the fourth term is 𝑎 four and so
on. One thing we could do is take a term
out of the summation. For example, we could remove 𝑎 one
from the summation and then start the series at 𝑛 equals two. So this series will be 𝑎 two add 𝑎
three add 𝑎 four and so on. So we get the same result.
We could, of course, take the first two
terms out instead and then start the series at 𝑛 equals three. This can come in handy when we want to
rewrite a series in order to use a particular theorem.
