Summation Question (Properties)

[dogma]

I have a rather simple question, but my rusty brain needs a good, swift kick-start.

I start with:

$$\sum_{i=1}^k i$$

and substitute in $$i=k-j$$ to get:

$$\sum_{k-j=1}^k (k-j)$$

How do I get from this to the following?

$$\sum_{k-j=1}^k (k-j) \rightarrow \sum_{j=0}^{k-1} (k-j)$$

Thanks in advance for your help.

dogma

 

[arildno]

I have a rather simple question, but my rusty brain needs a good, swift kick-start.

I start with:

$$\sum_{i=1}^k i$$

and substitute in $$i=k-j$$ to get:

$$\sum_{k-j=1}^k (k-j)$$

How do I get from this to the following?

$$\sum_{k-j=1}^k (k-j) \rightarrow \sum_{j=0}^{k-1} (k-j)$$

Thanks in advance for your help.

dogma

You're in some confusion in how to interpret the summation limits.
Let's write it explicitly, to see how it follows:
$$\sum_{i\geq{1}}^{i\leq{k}}i=\sum_{(k-j)\geq{1}}^{(k-j)\leq{k}}(k-j)$$

Now, rearrange the inequalities in the last expression:
$$\sum_{(k-j)\geq{1}}^{(k-j)\leq{k}}(k-j)=\sum_{(k-1)\geq{j}}^{0\leq{j}}(k-j)$$
Which in standard notation is nothing else than:
$$\sum_{j\geq{0}}^{j\leq(k-1)}(k-j)=\sum_{j=0}^{k-1}(k-j)$$

 

[dogma]

Thank you!

I completely understand now. I just need a good, swift kick.

Thanks again and take care!

dogma