Splitting sum using index notation

Question

Using index notation a sum $S=\sum_{i=1}^{N}a_i b_i$ can be written without the summation symbol since $i$ is a repeated index. Is it possible to write the sum in two terms

$$S=a_1 b_1 + \sum_{i=2}^{N}a_i b_i$$

in the same way (by using a Kronecker delta or other known symbols). If so, how?

@DanielY Why do you split a sum? Because you want special terms out Why do you use index notation? To avoid having tons of summation symbols everywhere. — user877329, Sep 16 '13 at 10:36
@DanielY Yes, but what if I do not like to write summation everywhere but just $a_i b_i=...$ — user877329, Sep 16 '13 at 14:36

score 1 · Accepted Answer · answered Nov 09 '13 at 05:57

In principle, you could write $$a_ib_i=a_ib_i \delta_{i1}+a_ib_i(1-\delta_{i1})$$ But please don't do this. For one thing, I never saw Einstein convention applied when an index appeared more than twice. For another, the convention is just a bad idea from the beginning [/opinion].

Splitting sum using index notation

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