Abs Series

Abs Series

(a) Find the series’ radius and interval of convergence. (b) For what values of x does the series converge abs?

infinitySigma n=0 (2x)^n
(a) Find the series’ radius and interval of convergence.
Radius?
lower bound?
upper bound?
(b) For what values of x does the series converge absolutely?
lower value?
upper value?

A) Use the ratio test. This test only applies when the terms are positive, so let’s look at the ratio of the moduli.

So |(n+1)st term|/|nth term|=|(2x)^(n+1)|/|(2x)^n|=2|x|.

Thus, when |x|<1/2, the series is convergent. Thus, the radius of convergence is R=1/2.

The interval of convergence is (-1/2, 1/2).

B) As far as I can tell, the absolute convergence should be the same as the regular convergence.

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Athletic Abs $11.16 Develop core strength, power and a rock hard mid section to help drive your athletic performance to a new level. This book combines a progressive series of abdominal strengthening exercises aimed at improving posture, body awareness, core strength, moto

Leisa Hart: Sexy Abs $12.71 Leisa Hart: Sexy ABS features the workout guru leading the viewer through a series of exercises designed to flatten, tone, and improve the appearance of one’s stomach. ~ Perry Seibert, Rovi

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Ripped Abs and No More Back Pain Series – Part IV

Interview With Jim Smith, CSCS – Author of Combat Core

I recently had a chance to sit down with Jim Smith, CSCS of the Diesel Crew and the author of Combat Core (replace Combat Core with your hoplink). I was able to get the low down on his new product and talk to him about what “real” core strength is all about.

[Jedd Johnson, CSCS] Question: Jim, First off, thanks for the interview. What do you think is the biggest mistake most trainers make when trying to develop core strength?

[JS] Most trainers focus on what I have dubbed building strength of movement patterns. What they fail to realize is that this is only one piece of the total puzzle. Building strength in the gym with movements like leg lifts, sit-ups, reverse sit-ups and so on…is a compliment to a bigger, more comprehensive core strength program. There are other criteria that make up the rest of the pyramid that I have established in Combat Core (replace Combat Core with your hoplink).

[Jedd Johnson, CSCS] Question: What, in your opinion, is the biggest myth concerning abdominal programs?

[JS] For trainers, I would point to my previous response. For the general public and even athletes, I would say that they believe that “more is better.” They believe, if they do 1000 crunches each workout, they will get ripped abs. Of course, the real answer is that being able to display a sick set of abs is the direct result of low body fat levels. If you want abs, you better get the fat off that is covering them.

[Jedd Johnson, CSCS] Question: How does core strength affect back pain and posture?

[JS] Your abdominals and back musculature work together to stabilize and protect the spine, hips and pelvis. If any of these muscle groups (and surrounding structures) are weak, posture is affected and sometimes the muscles (groups) become inhibited which causes the secondary movers to become overactive or on-tension. This will inevitably lead to injury and poor performance. Building torso strength by incorporating compound exercises that activate many muscle groups at the same time, teaches the lifter or athlete to move their body as a single, coordinated unit. Isolated exercises tend to lead to imbalances if used too much.

[Jedd Johnson, CSCS] Question: How has your abdominal training strategies changed over the years?

[JS] I used to think that by throwing in a couple sets of sit-ups or leg raises at the end of the workout was enough torso strengthening work. But over the years as I have gained experience and continued to study performance, I have developed a new, more comprehensive training model specific to athletes. The same attention and effort that you put in to planning your primary training sessions, you must also spend on designing your core training strategies.

About the Author

Jim Smith is a Certified Strength & Conditioning Specialist who writes for Men’s Fitness and the Elite Q/A Staff. Jim has been involved in strength training as a performance enhancement specialist for over 8 years and has worked with athletes from various sports who compete at various levels. He has published articles about his unique training style and innovative methods for many prominent strength and fitness related sites. He is also the authored of three renowned strength manuals. For more innovative training solutions, visit www.CombatCoreStrength.com.

Abs Workout | Lower Abs Series | Dr. Tim Ramirez