can someone do this step by step?

[fallow estuary]

I understand roughly where this comes from (the power series definition of the exponential), but I am having trouble manipulating the appropriate sums to produce these two equalities. can someone do it step by step? not asking a conceptual question here

[hybrid moat]

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[fallow estuary]

im also not sure where this formula comes from

[proud tundra]

$=\frac{\sum_{i=0}^\infty\frac{(nt)^i}{i!}-1}{t}=\frac{1}{t}\sum_{i=1}^\infty\frac{(nt)^i}{i!}=\sum_{i=1}^\infty\frac{n^it^{i-1}}{i!}$

[dusty valve]

Omegabet_

[proud tundra]

then they just reindexed

the latter is just the Cauchy product formula, you can find a proof of it online

but it's just multiplying 'infinitely long polynomials'

[fallow estuary]

very good cool

+close