# A not-hard-but-annoying thing

A couple times in class today I referred to something as not hard, but annoying. Here’s an example of the kind of thing I’m talking about:

Write the series $\sum_{n=20}^\infty (2n+1)$ as an infinite series starting at $n=0$.

Notice that $\sum_{n=20}^\infty (2n+1) = 2(20)+1 \; + \; 2(21)+1 \; + \; 2(22)+1 \; + \dots$. In particular, we want our $n=0$ term to equal $2(20)+1$, we want our $n=1$ term to equal $2(21)+1$, and so on. So it might be helpful to view our series as $2(20+0)+1 \; + \; 2(20+1)+1 \; + \; 2(20+2)+1 \; + \dots$. This would make general $n$ term $2(20+n)+1$.

So the series $\sum_{n=20}^\infty (2n+1)$ can be rewritten as $\sum_{n=0}^\infty (2(20+n)+1)$.

This actually isn’t brand new. Remember that if you have a function $f(x)$ and you want to graph a new function $g(x) = f(20+x)$, you just translate the graph of $f(x)$ to the left $20$ units. This also means that $g(0) = f(20), g(1)=f(21), g(2)=f(22)$ and so on.

My own brain knows that these translations are possible, but it just can’t remember whether it’d have to add or subtract 20 in the above example, so it has to work harder than it wants to to figure out that it should add 20. This is why I called it “not hard, but annoying.” (It might look hard the first time you see it, but after you’ve worked through enough examples like the one above, you’ll find it’s actually easy and hopefully not too annoying for your brain.)

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