Lb Versa
Lb Versa
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 Stamina Versa Bell 36 lb Adjustable Kettle Kettlebell NEW 05 3036 FREE SHIPPING  US $149.99
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 Stamina 36 Lb Adjustable Kettle Versa Bell 05 3036 US $116.99
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 Stamina 36 lb Adjustable Kettle Versa Bell 05 3036 US $125.00
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 Stamina 50 lb Versa Bell II Adjustable Dumbbell US $100.00
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 Stamina 50 Pound Versa Bell II Adjustable Dumbbell Pair US $357.79
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 Stamina 36 Pound Adjustable Kettle Versa Bell Workout Exercise Equipment US $140.45
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 Stamina 25 lb Versa Bell II pair US $192.95
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 Stamina 36 lb Adjustable Kettle Versa Bell NEW US $155.95
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 2 Stamina 50lb Adjustable Versa Bell Dumbbells 05 2155P US $358.00
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 Pair of Stamina 50 lb Versa Bell II Dumbbells 05 2150P US $332.00
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 One Stamina 50 lb VersaBell II Pro Dumbbell 05 2155 US $177.00
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 NEW 2 Stamina 25 lb Versa Bell II Dumbbells 05 2125P US $179.00
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 Stamina 36 lb Adjustable Kettle Versa Bell 16 36 lbs US $145.00
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 Stamina 50 lb Versa Bell II Dumbbell US $219.00
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 Stamina 50 lb Versa Bell II pair w Stand 05 2150K US $560.00
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 Stamina 25 lb Versa Bell II pair w Stand 05 2125K US $360.00
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 Stamina 25 lb Versa Bell Adjustable Dumbbell US $119.00
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 Stamina 25 lb Versa Bell II Dumbbell Pair US $219.00
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 Stamina 50 lb Versa Bell II Adjustable Dumbbell Pair US $439.00
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 Stamina Versa Bell II Dumbbell 25 lb US $89.99
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 Stamina Versa Bell II Dumbbell 50 lb US $149.99
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 Stamina 50 pound Versa Bell II Adjustable Dumbbells 2 US $294.99
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 Stamina 50 lb Versa Bell II Adjustable Dumbbell 05 2150 US $159.00
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 Stamina 36 lb Adjustable Kettle US $149.00
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 Stamina 50 LB Versa Bell II Pro Single Dumbbell NEW US $178.99
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 Stamina 36 Lb Adjustable Kettle Versa Bell NEW US $172.50
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 36Lb Adjustable Kettle Bell Strength training home gym workout 4lb increments NW US $168.95
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 2 Stamina® 50 lb Versa Bell II Pro Dumbbells 05 2155P US $358.00
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 Pair of Stamina® 25 lb Versa Bell® II Dumbbells 05 2125 US $186.00
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 One Stamina 50 lb Versa Bell II Pro Dumbbell 05 2155 US $177.00
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 Stamina® 50 Lb Versa Bell® Ii Dumbbell Pro 05 2155A US $159.99
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 Stamina® 50 Lb Versa Bell® II Adjustable Dumbbell Pair US $299.99
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 25 Pound Adjustable Dumbbell Pair Set Weights Gym NEW US $199.99
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 Stamina Versa Bell II 50lb Adjustable Dumbbell NEW 05 2150 US $149.99
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 Stamina Versa Bell II 50lb Adjustable Dumbbell Set Pair NEW 05 2150P US $289.00
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 Stamina® 25 Lb Versa Bell® II Dumbbell 05 2125A US $89.99
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 STAMINA 50 LB VERSA BELL II ADJUSTABLE DUMBELL US $149.00
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 STAMINA 25 LB VERSA BELL II ADJUSTABLE DUMBELL US $99.98
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 Stamina Kettle Versa Bell 05 3036 US $116.99
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 Stamina Versa Bell II Dumbbells Pair Up to 25 lbs 05 2125P US $134.99
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 Stamina® 50 Lb Versa Bell® II Pro Adj Dumbbell Pair US $299.99
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 Stamina® 36 Lb Adjustable Kettle Versa Bell® 05 3036 US $134.99
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 Stamina 05 3036 Adjustable Kettle Versa Bell 16 36 lbs US $149.99
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 Stamina 50lb Versa Bell Pro Adj Weight Dumbbell pair US $315.99
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 Stamina 25lb Versa Bell II Adj Weight Dumbbell Pair US $199.99
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 Stamina 50lb Versa Bell II Adj Weight Dumbbell Pair US $289.99
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Infinite series -- a hypothetical problem?
Consider the hypothetical problem illustrated in the figure:
Starting with a ball of radius 1 foot, a person stacks balls vertically such that if r_k is the radius of the kth ball, then r_(n+1) = (r_n)*sqrt[n/(n+1)] for each positive integer n.
(a) Show that the height of the stack can be made arbitarily large.
(b) If the balls are made of a material that weights 1 lb/ft^3, show
that the total weight of the stack is always less than 4pi pounds.
My difficulty is to find a closed form formula for r_n. Here are my thoughts:
I need to find a closed form formula for the radius, in order to do
part (b), which is summing the infinite series. I know if I have the closed form formula for radius, I cube and then multiply it by 4pi/3 to get the volume V_n and set V_n -> oo (infinity).
What are some general techniques for converting a recursive formula to a closed form formula, and vice versa? Any help will be appreciated.
Well, it's fairly obvious that r_n = 1/√n. The total height of the stack is the sum of the diameters, which is [n=1, ∞]∑2/√n, which diverges. Conversely, the mass of each ball is 4π/3 (r_n)³, so the total mass of the stack is 4π/3 [n=1, ∞]∑1/n^(3/2). This series converges by the integral test -- indeed, since [n-1, n]∫1/x^(3/2) dx > 1/n^(3/2), we have that:
[n=1, ∞]∑1/n^(3/2)
= 1 + [n=2, ∞]∑1/n^(3/2)
≤ 1 + [1, ∞]∫1/x^(3/2) dx
= 1 + (-2x^(-1/2))|[1, ∞]
= 1 + 2
= 3
So we have 4π/3 [n=1, ∞]∑1/n^(3/2) ≤ 4π. And we are done.
As far as general techniques for converting from a recursive definition to a closed form, this amounts to solving a difference equation. In most cases, you will just have to make intelligent guesses as to the form of the solution and then solve for the undetermined coefficients. In this case, though, you at least have the considerable advantage of being able to write out the first few r_n and look for a pattern, which was really obvious in this case.
The opposite problem, converting a closed-form expression into a recursive formula, is trivial. Given a closed-form expression x_n = f(n), simply write x_(n+1) = (f(n+1) - f(n)) + x_n or x_(n+1) = f(n+1)/f(n) * x_n, and voila -- instant recursive definition.