Would the sum of an infinite series be finite if there were gaps between each numbers that increased until reaching infinity? If you were to have a series:
1+1+2+4+7+11+16+22...
Where the gap between the numbers increased by one every time, wouldn't this gap eventually become infinite? Once it was, what would happen? Would it be possible to sum this series because although it is infinite, it has the infinite gap as well, so would this infinite gap stop the infinite series? And if it did, would the infinite series even be considered infinite, although it would have to be infinite in order for the gap to reach infinity.
 A: No. At any point the gap is finite. The gap will never become infinite.
A: Suppose there are 2 infinite ladders which you & I have to climb. You climb 1 step and shout, "I have reached my first position!" Now you climb 2 more steps and shout "I have reached my third location!" Then you climb 3 more steps and shout "I have reached my sixth location!" and so on..
Doing so, you generate a series 1,3,6,10... which corresponds to the step of ladder you are on.
I hear all your shouts, forever and climb on my ladder steps which are equal to the location number which you shout .You wait till I have climbed the steps, and then we climb again. I And you climb , forever.
Hence, given enough time, I will hear you say any number that exist in the number system, no matter however large.
The point where I am on the ladder is reached by adding all terms of the series you shout. Which then implies that the series can be summed. No matter how far along the ladder you go. 
The crux is that "gap increases to infinity " means no matter how large a number you give me , I can show you a point in my series after which the gap will always be larger than the number you gave me. It does not mean gap becomes infinity.(In our example, it cannot be that the number of steps that we have to take at any step of the above process is infinite.
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