Alternative account: User:Beneficii
Basic definition of a sum[edit]
Where :
Where :
Shifting of starting and ending indices[edit]
Proof of the equality of the shifting of indices:[edit]
Smaller summation notation[edit]
Combinations proof (used in below proof)[edit]
Proof by mathematical induction of the recursive geometric series (uses recursive summation notation)[edit]
Definition[edit]
Base case (and some specific examples)[edit]
Inductive step[edit]
Shifting of starting and ending indices (see above for proof):
See combinations proof above:
Shifting of starting and ending indices (see above for proof):
Adding case k=0 to the summation, means that the same must be subtracted from the summation:
Terms cancel out.
Q.E.D.
A general formula for recursive summation series[edit]
First proof, used in second proof[edit]
One method[edit]
Inductive method[edit]
Second proof, this one for the general formula for recursive summation series[edit]
Miscellaneous items (some valid, some not)[edit]
これ、ちょっとちがうね。
これもちがう。