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But the Bush administration’s shift toward war with Iraq left the Western powers without enough resources on the ground, so by 2006 the Taliban had regrouped into a serious military threat.

Mr. Obama, as a candidate, promised to refocus America’s resources on Afghanistan. But by the time Mr. Obama, as president, sent 33,000 more troops to Afghanistan in December 2009, years of war in Iraq and Afghanistan had drained Western resources and sapped resolve to build a viable Afghan state.

And over time, his administration has grown weary of trying to tackle Afghanistan’s seemingly intractable problems of poverty and corruption. The American people have grown weary, too.

While most Americans are sympathetic to the plight of the Afghan people, they have become deeply skeptical of President Hamid Karzai’s willingness to tackle corruption and political patronage and the coalition’s chances of “budging a medieval society” into the modern world, said Ann Marlowe, a visiting fellow at the Hudson Institute, a policy research organization in Washington.

“With millions of veterans home and talking with their families and friends … some knowledge of just how hard this is has percolated down,” said Ms. Marlowe, who has traveled to Afghanistan many times.

It has also been hard to show progress on the battlefield.

World War II had its Normandy, Vietnam its Tet Offensive and Iraq its Battle of Fallujah. Afghanistan is a grinding slough in villages and remote valleys where success if measured in increments.

The Afghan war transformed into a series of small, often vicious and intense fights scattered across a country almost as large as Texas.

In July, 40 U.S. service members died in Afghanistan in the deadliest month for American troops so far this year. At least 31 have been killed this month — seven when a helicopter crashed during a firefight with insurgents in what was one of the deadliest air disasters of the war. Ten others were gunned down in attacks from members of the Afghan security forces — either disgruntled turncoats or Taliban infiltrators.

Many argue that bin Laden’s death justifies a quick U.S. exit from Afghanistan. Others say it’s important to stay longer to shore up the Afghan security forces and help build the government so that it can stand on its own. An unstable Afghanistan could again offer sanctuary to militants like al Qaeda who want to harm American and its allies, they say.

“Those of us who have been at this for a long time continue to think that it’s important, and that we have a chance now of a path forward with a long-term perspective that will produce the results,” said James Cunningham, the new U.S. ambassador to Afghanistan.

The U.S.-led coalition’s combat mission will wind down in the next few years, leading up to the end of 2014, when most international troops will have left or moved into support roles.

Military analysts say the U.S. envisions a post-2014 force of perhaps 20,000 to hunt terrorists, train Afghan forces, and keep an eye on neighboring Iran and other regional powerhouse nations.

Americans aren’t likely to know the number until later this year. But will anyone other than families of service personnel take note?

“I have heard others say that the danger that their spouses or children are serving in is just simply not being cared about,” said Fred Wellman, a 22-year Army veteran who did three tours in Iraq. “I think a lot of veterans feel it is just forgotten.”

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