Hedge fund guru Carl Icahn tweeted today that he has invested $500 million more in Apple (AAPL -0.60%) stock. This brings his total investment to a whopping $3.6 billion -- or about 0.7% of the $500 billion company.

This announcement follows his tweet yesterday of a position that "crossed the $3 billion mark."

Along with his announcement of a larger position, he unveiled a seven-page letter to Apple shareholders making an in-depth case for Icahn's previously proposed larger share repurchase program. Apple's board has advised investors to vote against Icahn's proposal. Below I've summarized the key points in the seven-page latter, including Icahn's case for an Apple television.

iPhone 5s.

Apple is a "no brainer"
Icahn continues to insist that Apple stock is priced far too conservatively. To illustrate just how cheap Apple stock is, he points to the S&P 500's considerable premium to earnings compared to Apple's current valuation. By his calculations, if Apple traded at the same premium to earnings as the S&P 500, Apple shares would trade at $840 -- more than 50% higher than Apple's price at the time of this writing.

He says that over the course of his long investing career his best-performing investments come from opportunities he calls "no brainers." Apple is one of those opportunities, he argues. Reciting a few recent examples of no-brainers, he points to some of his recent winning stock picks, including Netflix, Forest Labs, and Herbalife -- all up more than 50% over the past 12 months.

It's the board's fiduciary responsibility to increase its buyback
His proposal is asking for $50 billion more in repurchases in 2014 alone and he believes Apple could easily handle such a boost to its buyback program. Just look at Apple's growing cash hoard, he argues. Not only does it have $130 billion of net cash, but analysts expect the company to earn about $40 billion in net income next year, he explains.

Given the conservative price of Apple shares and the company's massive cash hoard, Icahn believes "that it is the responsibility of the Board, on behalf of the company's shareholders, to take advantage of such a large and unmistakable opportunity."

Apple will continue to grow
But what about the naysayers who say Apple has ran out of opportunity for growth or, even worse, will soon see earnings decline?

There's plenty of opportunity for growth, Icahn argues. His argument for further growth is founded on the three following points:

  • The smartphone and tablet industry is expected to grow at 15% annually from 2013 through 2017, according to IDC.
  • Continuing loyalty to Apple's products suggests competitors won't erode Apple's pricing power.
  • New products in new categories will provide catalysts for growth

25 million new Apple ultra-high-definition televisions?
Since Icahn says he is not "not aware of a single Wall Street analyst who includes 'new products in new categories' or services in any of their financial projections,'" he took a stab at playing analyst himself.

If Apple introduces an ultra-high-definition television at $1,600 per unit, Icahn predicts Apple could sell 25 million units. The impact on Apple's business? "At a gross margin of 37.7%, which would be consistent with that of the overall company, such a debut would add $40 billion of revenues and $15 billion to operating income annually."

Icahn argues the opportunity for Apple to enter the television market is obvious, citing a narrowing price gap between high-definition televisions and ultra-high-definition technology and Apple's opportunity to capitalize more rapidly on 4K content than cable providers by debuting 4K digital movies in the iTunes store.

There are more opportunities beyond ultra-high-definition for Apple to find new growth markets, however, he says. Others include wearables and a payments service.

Should investors listen?
Icahn's letter certainly shouldn't alter investors' investment theses for Apple stock. But it does highlight some important points that are worth further contemplation.

What do you think? Is Apple being too conservative by arguing against Icahn's proposal for a larger repurchase program?