Jump to content
  • Advertisement
Sign in to follow this  

Summation: finding the upper bound algebraically

This topic is 4074 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

If you intended to correct an error in the post then please contact us.

Recommended Posts

Hello, Recently, I've encountered a rather unusual (for me) problem. Here's what it looks like: What is the minimum value of x for which this relationship is true? N ≤ A * Σ( (4b/5)^n, n=0, x-1 ) I need to find the minimum value of x, but notice that the x is found in the expression for the upper bound of the sum. I've never dealt with that before. Is there a way to solve this algebraically? The context of the problem gives some additional indications for the other variables. First, all variables are whole numbers. Second, the value of b is an integer between 0 and 8 (but no more than 8). If there is no way to solve this symbolically, then I suppose that I will be content with an estimate. However, my attempt to find a suitable estimate led me to another mathematical dead end. This year in my calculus class, we learned the integral test: Σ(ak, k=1, ∞) ≤ a1 + ∫(a(x)dx, 1, ∞) Hence, this must be true (right?): N ≤ A + ∫( ( 4b/5)^t, 0, x-1 ) (Because the original summation was a geometric sequence in the form ar^n where a = A, the first term a1 = A). From here, I antidifferentiated and substituted. Then I multiplied the last term of the resulting relation by (4b/5)/(4b/5) to get this: N ≤ A + ((4b/5)x-(4b/5)2)/((4b/5)*ln(4b/5)) I'm pretty sure my work is right... but where do I go from here? I predict that when I solve this equation, the result will be in the form of: x ≥ some very complex expression From that point, would it be admissible to simply plug in the known values of N, A, and b and round up to the nearest whole number (the "ceiling")? Would that yield a valid estimate? Surely there is a simpler way to solve (or approximate) this? Please help!

Share this post


Link to post
Share on other sites
Advertisement
I suppose you are referring to this (from the linked article):



Thanks for the help. I'll try to work with it and see what happens.

Share this post


Link to post
Share on other sites
Sign in to follow this  

  • Advertisement
×

Important Information

By using GameDev.net, you agree to our community Guidelines, Terms of Use, and Privacy Policy.

Participate in the game development conversation and more when you create an account on GameDev.net!

Sign me up!