To find the sum of the given series, we need to add all the terms together.
(1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/2009 - 1/2010)
We can simplify each term by finding the common denominator.
1 - 1/2 = 2/2 - 1/2 = 1/2
1/2 - 1/3 = 3/6 - 2/6 = 1/6
1/3 - 1/4 = 4/12 - 3/12 = 1/12
We can observe that each term follows this pattern - the denominator of the second fraction becomes the denominator of the first fraction in the next term.
So, the series can be written as:
1/2 + 1/6 + 1/12 + ... + 1/2009
To find the sum of this series, we need to find the common denominator of all the fractions.
The common denominator of 2, 6, 12, ..., 2009 will be the least common multiple (LCM) of these numbers.
Calculating the LCM of these numbers is a bit lengthy. Instead, we can find the LCM of 2, 3, 4, ..., 2010, and then divide by the LCM of 2, 3, 4, ..., 2009.
LCM(2, 3, 4, ..., 2010) / LCM(2, 3, 4, ..., 2009)
= 2010 / 2
= 1005
So, the common denominator is 1005.
To add the fractions, we need to express them with the common denominator:
1/2 = (1/2) * (1005/1005) = 1005/2010
1/6 = (1/6) * (1005/1005) = 167.5/2010
1/12 = (1/12) * (1005/1005) = 83.75/2010
Now we can add:
1005/2010 + 167.5/2010 + 83.75/2010 + ... + 1/2009
We can observe that the denominators of the fractions form an arithmetic sequence, and the numerators follow the same pattern.
Using the formula for the sum of an arithmetic sequence:
Sum = (first term + last term) * number of terms / 2
In this case, the first term is 1005/2010, the last term is 1/2009, and the number of terms is 2010.
Sum = (1005/2010 + 1/2009) * 2010/2
Sum = (1005/2010 + 1/2009) * 1005
Sum = (1005 * 2009 + 1 * 2010) / 2
Sum = (2019955 + 2010) / 2
Sum = 2021965 / 2
Sum = 1010982.5
Therefore, the sum of the given series is 1010982.5.
他亲自组织政府相关部门，前往现场调研项目进展情况，并召集相关部门人员进行项目论证，制定方案精准、措施得力，他要求相关部门迅速协调解决建设资金问题，确保项目有序进行，尽早完工投入使用。， 合肥场景公司负责人介绍，本次场景创新大赛是合肥推深做实应用场景创新工作推出的又一重磅举措，通过科技赋能民生、场景赋能科技，遴选出一批能应用、可复制的产品，落地到城市生活当中，打通科学技术与民众生活的壁垒，让科创走进民生，惠及民生。
（总台记者 李长皓）， “你先自己挑着，我去招呼别的客人，你决定了和我说。
当前，Stellantis在中国实行轻资产运营模式。， 阿森纳其实也受到了不小影响，上赛季枪手就已经展现出了冠军潜力，今夏又引进了赖斯、哈弗茨等多位球员，成绩不如上赛季同期的一个重要原因就在于本赛季要踢欧冠——首轮小组赛击败埃因霍温后主场平热刺，第三轮客场击败塞维利亚前平了切尔西，第四轮主场战胜塞维利亚前输给纽卡斯尔，丢分的四场比赛中只有2比2平富勒姆和欧冠没有关系。
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