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In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑n=1∞n0{displaystyle sum _{n=1}^{infty }n^{0}}, ∑n=1∞1n{displaystyle sum _{n=1}^{infty }1^{n}}, or simply ∑n=1∞1{displaystyle sum _{n=1}^{infty }1}, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1ⁿ can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line

∑n=1∞1=+∞,{displaystyle sum _{n=1}^{infty }1=+infty ,}

since its sequence of partial sums increases monotonically without bound.

Monosnap 3 5 8. Where the sum of n⁰ occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function

ζ(s)=∑n=1∞1ns=11−21−s∑n=1∞(−1)n+1ns,{displaystyle zeta (s)=sum _{n=1}^{infty }{frac {1}{n^{s}}}={frac {1}{1-2^{1-s}}}sum _{n=1}^{infty }{frac {(-1)^{n+1}}{n^{s}}},}

The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function,

ζ(s)=2sπs−1sin⁡(πs2)Γ(1−s)ζ(1−s),{displaystyle zeta (s)=2^{s}pi ^{s-1} sin left({frac {pi s}{2}}right) Gamma (1-s) zeta (1-s)!,}

Using this one gets (given that Γ(1) = 1),

since its sequence of partial sums increases monotonically without bound.

Monosnap 3 5 8. Where the sum of n⁰ occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function

ζ(s)=∑n=1∞1ns=11−21−s∑n=1∞(−1)n+1ns,{displaystyle zeta (s)=sum _{n=1}^{infty }{frac {1}{n^{s}}}={frac {1}{1-2^{1-s}}}sum _{n=1}^{infty }{frac {(-1)^{n+1}}{n^{s}}},}

The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function,

ζ(s)=2sπs−1sin⁡(πs2)Γ(1−s)ζ(1−s),{displaystyle zeta (s)=2^{s}pi ^{s-1} sin left({frac {pi s}{2}}right) Gamma (1-s) zeta (1-s)!,}

Using this one gets (given that Γ(1) = 1),

ζ(0)=1πlims→0sin⁡(πs2)ζ(1−s)=1πlims→0(πs2−π3s348+...)(−1s+...)=−12{displaystyle zeta (0)={frac {1}{pi }}lim _{srightarrow 0} sin left({frac {pi s}{2}}right) zeta (1-s)={frac {1}{pi }}lim _{srightarrow 0} left({frac {pi s}{2}}-{frac {pi ^{3}s^{3}}{48}}+.right) left(-{frac {1}{s}}+.right)=-{frac {1}{2}}}

where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2.

Emilio Elizalde presents a comment from others about the series:

In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.^[2]

See also[edit]

Notes[edit]

1. ^Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
2. ^Elizalde, Emilio (2004). 'Cosmology: Techniques and Applications'. Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc..9076E.

External links[edit]

• OEISsequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)

1.1.1.1 is a free Domain Name System (DNS) service. The public DNS service and servers are maintained and owned by Cloudflare in partnership with APNIC.^[1] The service functions as a recursive name server providing domain name resolution for any host on the Internet. The service was announced on April 1, 2018,^[2] and is claimed by Cloudflare to be 'the Internet's fastest, privacy-first consumer DNS service'.^[3] On November 11, 2018, Cloudflare announced a mobile application of their 1.1.1.1 service for Android and iOS.^[4] On September 25, 2019, Cloudflare released WARP, an upgraded version of their original 1.1.1.1 mobile application.^[5]

Backgrounds 7 1st. According to DNSPerf, 1.1.1.1 is the world's fastest recursive DNS resolver, beating other popular resolvers such as Google's Public DNS resolver. ^[6]

Service[edit]

The 1.1.1.1 DNS service operates recursive name servers for public use at the four following IP addresses.^[7] The addresses are mapped to the nearest operational server by anycast routing.^[8] The DNS service is also available for Tor clients.^[9] Users can set up the service by manually changing their DNS resolvers to the IP addresses below. Mobile users on both Android and iPhone have the alternative of downloading the 1.1.1.1 mobile application, which automatically configures the DNS resolvers on the device.^[10]

Technology[edit]

Potato 24. 1.1.1.1 is successful as a recursive DNS resolver because of Cloudflare's vast network. Cloudflare runs an authoritative DNS resolver with a network of over 20 million Internet properties.^[12] With the recursor and the resolver on the same network, DNS queries can be answered faster than the existing resolvers.

With the release of the 1.1.1.1 mobile application in November 2018, Cloudflare added the ability for users to encrypt their DNS queries over HTTPS (DoH) or TLS (DoT).^[13] Later on, WARP was implemented using a new protocol, WireGuard, which acts as a hyper-efficient VPN tunnel.^[14]

Exposure of abuse[edit]

Technology websites noted that by using 1.1.1.1 as the IP address for its service, Cloudflare exposed misconfigurations in existing setups that violated Internet standards (such as RFC1918). 1.1.1.1 was not a reserved IP address, yet was abused by many existing routers (mostly those sold by Cisco Systems) and companies for hosting login pages to private networks, exit pages or other purposes, rendering the proper routing of 1.1.1.1 impossible on those systems.^[15][16] Additionally, 1.1.1.1 is blocked on many networks and by multiple ISPs because the simplicity of the address means that it was previously often used inappropriately for testing purposes and not legitimate use.^[15] These previous uses have led to a huge influx of garbage data to Cloudflare's servers.^[16]

Cleanup of 1.1.1.1 and 1.0.0.1[edit]

The 1.0.0.0/8 IP block was assigned in 2010 to APNIC;^[17] before this time it was unassigned space.^[18] An unassigned IP space, however is not the same as a reserved IP space for private use (called a reserved IP address).^[19] For example, AT&T has said it is working on fixing this issue^[20] within its CPE hardware.

Privacy Policy[edit]

1.1.1. https://gugumaxi287.weebly.com/cleanmydrive-2-1-8-manage-and-clean-external-drives.html. 1 was built as an alternative to the default DNS resolvers from Internet Service Providers (ISP). Since every internet query needs to go through a DNS resolver in order to translate a text-based web address to a numerical IP address, DNS resolvers hold a lot of data on its users. Owners of DNS services, such as an ISP, are able to track exactly what websites a user visits.^[6] Cloudflare's 1.1.1.1 markets itself as the antithesis to these traditional DNS resolvers. The service claims to adhere to the following privacy principles: ^[21]

1. Don't write user-identifiable log data to disk;
2. Will never sell browsing data or use it in any way to target users with advertising data;
3. The end user is not required to provide any personal information — not their name, phone number, or email address — in order to use the 1.1.1.1 App with Warp; and
4. Will regularly hire outside auditors to ensure the service is living up to these promises.

WARP[edit]

On April 1, 2019, Cloudflare announced they were planning to launch a VPN service called WARP which would be built into the 1.1.1.1 mobile app. The standard service would be provided for free, with a paid tier which includes additional features.^[22] The service was announced in April and millions of people signed up to get WARP.^[23] In the following months, Cloudflare did not release any updates or announcements on WARP, which led to frustrations and negative feedback.^[24] After much anticipation, Cloudflare released WARP to the public on September 25, 2019. ^[25][26]

WARP is an update to Cloudflare's existing 1.1.1.1 mobile application. People can use the application in 3 different modes: Usher 1 1 17.

• 1.1.1.1 uses Cloudflare's public DNS resolver to encrypt DNS queries over DoH and DoT.
• 1.1.1.1 with WARP allows users to encrypt all their mobile traffic instead of just the DNS queries.
• WARP+ is the paid premium tier of the service. Using Cloudflare's Argo Smart Routing, WARP+ routes your traffic through the 'Internet fast-lane', which makes websites load 30% faster on average.^[27]

See also[edit]

References[edit]

1. ^APNIC Labs enters into a research agreement with Cloudflare, Geoff Huston, APNIC Blog, April 2, 2018
2. ^Cloudflare launches 1.1.1.1 DNS service that will speed up your internet The Verge, April 1, 2018
3. ^Announcing 1.1.1.1: the fastest, privacy-first consumer DNS service, Official Cloudflare Blog
4. ^Cimpanu, Catalin. 'Cloudflare launches Android and iOS apps for its 1.1.1.1 service | ZDNet'. ZDNet.
5. ^'WARP is here (sorry it took so long)'. The Cloudflare Blog. September 25, 2019. Retrieved November 20, 2019.
6. ^ ^abProspectOne. 'DNS Performance'. www.dnsperf.com. Retrieved November 20, 2019.
7. ^Setting Up 1.1.1.1 Speed
8. ^Introducing DNS Resolver, 1.1.1.1 (not a joke) DNS resolver, 1.1.1.1, is served by Cloudflare's Global Anycast Network.
9. ^'Introducing DNS Resolver for Tor'. Cloudflare. June 5, 2018. Retrieved October 1, 2018.
10. ^1.1.1.1. '1.1.1.1 — The free app that makes your Internet faster'. 1.1.1.1. Retrieved November 22, 2019.CS1 maint: numeric names: authors list (link)
11. ^'Enable Private DNS with 1.1.1.1 on Android 9 Pie'. Cloudflare. August 16, 2018. Retrieved October 6, 2020.
12. ^'Managed DNS Provider | Cloud DNS Service'. Cloudflare. Retrieved November 22, 2019.
13. ^'Introducing Warp: Fixing Mobile Internet Performance and Security'. The Cloudflare Blog. April 1, 2019. Retrieved November 22, 2019.
14. ^'The Technical Challenges of Building Cloudflare WARP'. The Cloudflare Blog. September 25, 2019. Retrieved November 22, 2019.
15. ^ ^abCherry, Denny (April 5, 2018). '5 reasons Cloudflare's roll-out of 1.1.1.1 has been a disaster'. Tech Target. Archived from the original on April 26, 2018. Retrieved April 26, 2018.
16. ^ ^ab'1.1.1.1: Cloudflare's new DNS attracting 'gigabits per second' of rubbish'. ZDNet. April 4, 2018. Retrieved April 26, 2018.
17. ^'1/8 and 27/8 allocated to APNIC'. NANOG. January 21, 2010. Retrieved May 3, 2018.
18. ^List of assigned /8 IPv4 address blocks
19. ^Fixing reachability to 1.1.1.1, GLOBALLY!, by Marty Strong, April 10, 2018
20. ^Tweet by @billplein, 3 April 2018
21. ^'Introducing Warp: Fixing Mobile Internet Performance and Security'. The Cloudflare Blog. April 1, 2019. Retrieved November 22, 2019.
22. ^Khalid, Amrita (April 2, 2019). 'Cloudflare's privacy-focused DNS app adds a free VPN'. Engadget. Archived from the original on April 2, 2019. Retrieved April 2, 2019.
23. ^'WARP is here (sorry it took so long)'. The Cloudflare Blog. September 25, 2019. Retrieved November 22, 2019.
24. ^'r/CloudFlare - Cloudflare WARP????????'. reddit. Retrieved November 22, 2019.
25. ^Humphries, By Matthew; September 26, 2019 10:18AM EST; September 26, 2019. 'Cloudflare Finally Launches Warp, But It's Not a Mobile VPN'. PCMAG. Retrieved September 27, 2019.CS1 maint: numeric names: authors list (link)
26. ^Security, Paul Wagenseil 2019-09-26T20:13:55Z. 'WARP Promises Faster Speeds on Your Phone Without 5G, but Doesn't Quite Deliver Yet'. Tom's Guide. Retrieved September 27, 2019.
27. ^1.1.1.1. '1.1.1.1 — The free app that makes your Internet faster'. 1.1.1.1. Retrieved November 22, 2019.CS1 maint: numeric names: authors list (link)

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